Create a Perl library directory, and add the Graph module to it

Graph-0.84 from CPAN

22 files changed, 10912 insertions, 0 deletions

diff --git a/perllib/Graph.pm b/perllib/Graph.pm
new file mode 100644
index 00000000..3d1ad336
--- /dev/null
+++ b/perllib/Graph.pm
@@ -0,0 +1,3851 @@
+package Graph;
+
+use strict;
+
+BEGIN {
+    if (0) { # SET THIS TO ZERO FOR TESTING AND RELEASES!
+	$SIG{__DIE__ } = \&__carp_confess;
+	$SIG{__WARN__} = \&__carp_confess;
+    }
+    sub __carp_confess { require Carp; Carp::confess(@_) }
+}
+
+use Graph::AdjacencyMap qw(:flags :fields);
+
+use vars qw($VERSION);
+
+$VERSION = '0.84';
+
+require 5.006; # Weak references are absolutely required.
+
+use Graph::AdjacencyMap::Heavy;
+use Graph::AdjacencyMap::Light;
+use Graph::AdjacencyMap::Vertex;
+use Graph::UnionFind;
+use Graph::TransitiveClosure;
+use Graph::Traversal::DFS;
+use Graph::MSTHeapElem;
+use Graph::SPTHeapElem;
+use Graph::Undirected;
+
+use Heap071::Fibonacci;
+use List::Util qw(shuffle first);
+use Scalar::Util qw(weaken);
+
+sub _F () { 0 } # Flags.
+sub _G () { 1 } # Generation.
+sub _V () { 2 } # Vertices.
+sub _E () { 3 } # Edges.
+sub _A () { 4 } # Attributes.
+sub _U () { 5 } # Union-Find.
+
+my $Inf;
+
+BEGIN {
+    local $SIG{FPE}; 
+    eval { $Inf = exp(999) } ||
+	eval { $Inf = 9**9**9 } ||
+	    eval { $Inf = 1e+999 } ||
+		{ $Inf = 1e+99 };  # Close enough for most practical purposes.
+}
+
+sub Infinity () { $Inf }
+
+# Graphs are blessed array references.
+# - The first element contains the flags.
+# - The second element is the vertices.
+# - The third element is the edges.
+# - The fourth element is the attributes of the whole graph.
+# The defined flags for Graph are:
+# - _COMPAT02 for user API compatibility with the Graph 0.20xxx series.
+# The vertices are contained in either a "simplemap"
+# (if no hypervertices) or in a "map".
+# The edges are always in a "map".
+# The defined flags for maps are:
+# - _COUNT for countedness: more than one instance
+# - _HYPER for hyperness: a different number of "coordinates" than usual;
+#   expects one for vertices and two for edges
+# - _UNORD for unordered coordinates (a set): if _UNORD is not set
+#   the coordinates are assumed to be meaningfully ordered
+# - _UNIQ for unique coordinates: if set duplicates are removed,
+#   if not, duplicates are assumed to meaningful
+# - _UNORDUNIQ: just a union of _UNORD and UNIQ
+# Vertices are assumed to be _UNORDUNIQ; edges assume none of these flags.
+
+use Graph::Attribute array => _A, map => 'graph';
+
+sub _COMPAT02 () { 0x00000001 }
+
+sub stringify {
+    my $g = shift;
+    my $o = $g->is_undirected;
+    my $e = $o ? '=' : '-';
+    my @e =
+	map {
+	    my @v =
+		map {
+		    ref($_) eq 'ARRAY' ? "[" . join(" ", @$_) . "]" : "$_"
+		}
+	    @$_;
+	    join($e, $o ? sort { "$a" cmp "$b" } @v : @v) } $g->edges05;
+    my @s = sort { "$a" cmp "$b" } @e;
+    push @s, sort { "$a" cmp "$b" } $g->isolated_vertices;
+    join(",", @s);
+}
+
+sub eq {
+    "$_[0]" eq "$_[1]"
+}
+
+sub ne {
+    "$_[0]" ne "$_[1]"
+}
+
+use overload
+    '""' => \&stringify,
+    'eq' => \&eq,
+    'ne' => \≠
+
+sub _opt {
+    my ($opt, $flags, %flags) = @_;
+    while (my ($flag, $FLAG) = each %flags) {
+	if (exists $opt->{$flag}) {
+	    $$flags |= $FLAG if $opt->{$flag};
+	    delete $opt->{$flag};
+	}
+	if (exists $opt->{my $non = "non$flag"}) {
+	    $$flags &= ~$FLAG if $opt->{$non};
+	    delete $opt->{$non};
+	}
+    }
+}
+
+sub is_compat02 {
+    my ($g) = @_;
+    $g->[ _F ] & _COMPAT02;
+}
+
+*compat02 = \&is_compat02;
+
+sub has_union_find {
+    my ($g) = @_;
+    ($g->[ _F ] & _UNIONFIND) && defined $g->[ _U ];
+}
+
+sub _get_union_find {
+    my ($g) = @_;
+    $g->[ _U ];
+}
+
+sub _opt_get {
+    my ($opt, $key, $var) = @_;
+    if (exists $opt->{$key}) {
+	$$var = $opt->{$key};
+	delete $opt->{$key};
+    }
+}
+
+sub _opt_unknown {
+    my ($opt) = @_;
+    if (my @opt = keys %$opt) {
+	my $f = (caller(1))[3];
+	require Carp;
+	Carp::confess(sprintf
+		      "$f: Unknown option%s: @{[map { qq['$_'] } sort @opt]}",
+		      @opt > 1 ? 's' : '');
+    }
+}
+
+sub new {
+    my $class = shift;
+    my $gflags = 0;
+    my $vflags;
+    my $eflags;
+    my %opt = _get_options( \@_ );
+
+    if (ref $class && $class->isa('Graph')) {
+	no strict 'refs';
+        for my $c (qw(undirected refvertexed compat02
+                      hypervertexed countvertexed multivertexed
+                      hyperedged countedged multiedged omniedged)) {
+#            $opt{$c}++ if $class->$c; # 5.00504-incompatible
+	    if (&{"Graph::$c"}($class)) { $opt{$c}++ }
+        }
+#        $opt{unionfind}++ if $class->has_union_find; # 5.00504-incompatible
+	if (&{"Graph::has_union_find"}($class)) { $opt{unionfind}++ }
+    }
+
+    _opt_get(\%opt, undirected   => \$opt{omniedged});
+    _opt_get(\%opt, omnidirected => \$opt{omniedged});
+
+    if (exists $opt{directed}) {
+	$opt{omniedged} = !$opt{directed};
+	delete $opt{directed};
+    }
+
+    my $vnonomni =
+	$opt{nonomnivertexed} ||
+	    (exists $opt{omnivertexed} && !$opt{omnivertexed});
+    my $vnonuniq =
+	$opt{nonuniqvertexed} ||
+	    (exists $opt{uniqvertexed} && !$opt{uniqvertexed});
+
+    _opt(\%opt, \$vflags,
+	 countvertexed	=> _COUNT,
+	 multivertexed	=> _MULTI,
+	 hypervertexed	=> _HYPER,
+	 omnivertexed	=> _UNORD,
+	 uniqvertexed	=> _UNIQ,
+	 refvertexed	=> _REF,
+	);
+
+    _opt(\%opt, \$eflags,
+	 countedged	=> _COUNT,
+	 multiedged	=> _MULTI,
+	 hyperedged	=> _HYPER,
+	 omniedged	=> _UNORD,
+	 uniqedged	=> _UNIQ,
+	);
+
+    _opt(\%opt, \$gflags,
+	 compat02      => _COMPAT02,
+	 unionfind     => _UNIONFIND,
+	);
+
+    if (exists $opt{vertices_unsorted}) { # Graph 0.20103 compat.
+	my $unsorted = $opt{vertices_unsorted};
+	delete $opt{vertices_unsorted};
+	require Carp;
+	Carp::confess("Graph: vertices_unsorted must be true")
+	    unless $unsorted;
+    }
+
+    my @V;
+    if ($opt{vertices}) {
+	require Carp;
+	Carp::confess("Graph: vertices should be an array ref")
+	    unless ref $opt{vertices} eq 'ARRAY';
+	@V = @{ $opt{vertices} };
+	delete $opt{vertices};
+    }
+
+    my @E;
+    if ($opt{edges}) {
+	unless (ref $opt{edges} eq 'ARRAY') {
+	    require Carp;
+	    Carp::confess("Graph: edges should be an array ref of array refs");
+	}
+	@E = @{ $opt{edges} };
+	delete $opt{edges};
+    }
+
+    _opt_unknown(\%opt);
+
+    my $uflags;
+    if (defined $vflags) {
+	$uflags = $vflags;
+	$uflags |= _UNORD unless $vnonomni;
+	$uflags |= _UNIQ  unless $vnonuniq;
+    } else {
+	$uflags = _UNORDUNIQ;
+	$vflags = 0;
+    }
+
+    if (!($vflags & _HYPER) && ($vflags & _UNORDUNIQ)) {
+	my @but;
+	push @but, 'unordered' if ($vflags & _UNORD);
+	push @but, 'unique'    if ($vflags & _UNIQ);
+	require Carp;
+	Carp::confess(sprintf "Graph: not hypervertexed but %s",
+		      join(' and ', @but));
+    }
+
+    unless (defined $eflags) {
+	$eflags = ($gflags & _COMPAT02) ? _COUNT : 0;
+    }
+
+    if (!($vflags & _HYPER) && ($vflags & _UNIQ)) {
+	require Carp;
+	Carp::confess("Graph: not hypervertexed but uniqvertexed");
+    }
+
+    if (($vflags & _COUNT) && ($vflags & _MULTI)) {
+	require Carp;
+	Carp::confess("Graph: both countvertexed and multivertexed");
+    }
+
+    if (($eflags & _COUNT) && ($eflags & _MULTI)) {
+	require Carp;
+	Carp::confess("Graph: both countedged and multiedged");
+    }
+
+    my $g = bless [ ], ref $class || $class;
+
+    $g->[ _F ] = $gflags;
+    $g->[ _G ] = 0;
+    $g->[ _V ] = ($vflags & (_HYPER | _MULTI)) ?
+	Graph::AdjacencyMap::Heavy->_new($uflags, 1) :
+	    (($vflags & ~_UNORD) ?
+	     Graph::AdjacencyMap::Vertex->_new($uflags, 1) :
+	     Graph::AdjacencyMap::Light->_new($g, $uflags, 1));
+    $g->[ _E ] = (($vflags & _HYPER) || ($eflags & ~_UNORD)) ?
+	Graph::AdjacencyMap::Heavy->_new($eflags, 2) :
+	    Graph::AdjacencyMap::Light->_new($g, $eflags, 2);
+
+    $g->add_vertices(@V) if @V;
+
+    if (@E) {
+	for my $e (@E) {
+	    unless (ref $e eq 'ARRAY') {
+		require Carp;
+		Carp::confess("Graph: edges should be array refs");
+	    }
+	    $g->add_edge(@$e);
+	}
+    }
+
+    if (($gflags & _UNIONFIND)) {
+	$g->[ _U ] = Graph::UnionFind->new;
+    }
+
+    return $g;
+}
+
+sub countvertexed { $_[0]->[ _V ]->_is_COUNT }
+sub multivertexed { $_[0]->[ _V ]->_is_MULTI }
+sub hypervertexed { $_[0]->[ _V ]->_is_HYPER }
+sub omnivertexed  { $_[0]->[ _V ]->_is_UNORD }
+sub uniqvertexed  { $_[0]->[ _V ]->_is_UNIQ  }
+sub refvertexed   { $_[0]->[ _V ]->_is_REF   }
+
+sub countedged    { $_[0]->[ _E ]->_is_COUNT }
+sub multiedged    { $_[0]->[ _E ]->_is_MULTI }
+sub hyperedged    { $_[0]->[ _E ]->_is_HYPER }
+sub omniedged     { $_[0]->[ _E ]->_is_UNORD }
+sub uniqedged     { $_[0]->[ _E ]->_is_UNIQ  }
+
+*undirected   = \&omniedged;
+*omnidirected = \&omniedged;
+sub directed { ! $_[0]->[ _E ]->_is_UNORD }
+
+*is_directed      = \&directed;
+*is_undirected    = \&undirected;
+
+*is_countvertexed = \&countvertexed;
+*is_multivertexed = \&multivertexed;
+*is_hypervertexed = \&hypervertexed;
+*is_omnidirected  = \&omnidirected;
+*is_uniqvertexed  = \&uniqvertexed;
+*is_refvertexed   = \&refvertexed;
+
+*is_countedged    = \&countedged;
+*is_multiedged    = \&multiedged;
+*is_hyperedged    = \&hyperedged;
+*is_omniedged     = \&omniedged;
+*is_uniqedged     = \&uniqedged;
+
+sub _union_find_add_vertex {
+    my ($g, $v) = @_;
+    my $UF = $g->[ _U ];
+    $UF->add( $g->[ _V ]->_get_path_id( $v ) );
+}
+
+sub add_vertex {
+    my $g = shift;
+    if ($g->is_multivertexed) {
+	return $g->add_vertex_by_id(@_, _GEN_ID);
+    }
+    my @r;
+    if (@_ > 1) {
+	unless ($g->is_countvertexed || $g->is_hypervertexed) {
+	    require Carp;
+	    Carp::croak("Graph::add_vertex: use add_vertices for more than one vertex or use hypervertexed");
+	}
+	for my $v ( @_ ) {
+	    if (defined $v) {
+		$g->[ _V ]->set_path( $v ) unless $g->has_vertex( $v );
+	    } else {
+		require Carp;
+		Carp::croak("Graph::add_vertex: undef vertex");
+	    }
+	}
+    }
+    for my $v ( @_ ) {
+	unless (defined $v) {
+	    require Carp;
+	    Carp::croak("Graph::add_vertex: undef vertex");
+	}
+    }
+    $g->[ _V ]->set_path( @_ );
+    $g->[ _G ]++;
+    $g->_union_find_add_vertex( @_ ) if $g->has_union_find;
+    return $g;
+}
+
+sub has_vertex {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    return exists $V->[ _s ]->{ $_[0] } if ($V->[ _f ] & _LIGHT);
+    $V->has_path( @_ );
+}
+
+sub vertices05 {
+    my $g = shift;
+    my @v = $g->[ _V ]->paths( @_ );
+    if (wantarray) {
+	return $g->[ _V ]->_is_HYPER ?
+	    @v : map { ref $_ eq 'ARRAY' ? @$_ : $_ } @v;
+    } else {
+	return scalar @v;
+    }
+}
+
+sub vertices {
+    my $g = shift;
+    my @v = $g->vertices05;
+    if ($g->is_compat02) {
+        wantarray ? sort @v : scalar @v;
+    } else {
+	if ($g->is_multivertexed || $g->is_countvertexed) {
+	    if (wantarray) {
+		my @V;
+		for my $v ( @v ) {
+		    push @V, ($v) x $g->get_vertex_count($v);
+		}
+		return @V;
+	    } else {
+		my $V = 0;
+		for my $v ( @v ) {
+		    $V += $g->get_vertex_count($v);
+		}
+		return $V;
+	    }
+	} else {
+	    return @v;
+	}
+    }
+}
+
+*vertices_unsorted = \&vertices_unsorted; # Graph 0.20103 compat.
+
+sub unique_vertices {
+    my $g = shift;
+    my @v = $g->vertices05;
+    if ($g->is_compat02) {
+        wantarray ? sort @v : scalar @v;
+    } else {
+	return @v;
+    }
+}
+
+sub has_vertices {
+    my $g = shift;
+    scalar $g->[ _V ]->has_paths( @_ );
+}
+
+sub _add_edge {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    my @e;
+    if (($V->[ _f ]) & _LIGHT) {
+	for my $v ( @_ ) {
+	    $g->add_vertex( $v ) unless exists $V->[ _s ]->{ $v };
+	    push @e, $V->[ _s ]->{ $v };
+	}
+    } else {
+	my $h = $g->[ _V ]->_is_HYPER;
+	for my $v ( @_ ) {
+	    my @v = ref $v eq 'ARRAY' && $h ? @$v : $v;
+	    $g->add_vertex( @v ) unless $V->has_path( @v );
+	    push @e, $V->_get_path_id( @v );
+	}
+    }
+    return @e;
+}
+
+sub _union_find_add_edge {
+    my ($g, $u, $v) = @_;
+    $g->[ _U ]->union($u, $v);
+}
+
+sub add_edge {
+    my $g = shift;
+    if ($g->is_multiedged) {
+	unless (@_ == 2 || $g->is_hyperedged) {
+	    require Carp;
+	    Carp::croak("Graph::add_edge: use add_edges for more than one edge");
+	}
+	return $g->add_edge_by_id(@_, _GEN_ID);
+    }
+    unless (@_ == 2) {
+	unless ($g->is_hyperedged) {
+	    require Carp;
+	    Carp::croak("Graph::add_edge: graph is not hyperedged");
+	}
+    }
+    my @e = $g->_add_edge( @_ );
+    $g->[ _E ]->set_path( @e );
+    $g->[ _G ]++;
+    $g->_union_find_add_edge( @e ) if $g->has_union_find;
+    return $g;
+}
+
+sub _vertex_ids {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    my @e;
+    if (($V->[ _f ] & _LIGHT)) {
+	for my $v ( @_ ) {
+	    return () unless exists $V->[ _s ]->{ $v };
+	    push @e, $V->[ _s ]->{ $v };
+	}
+    } else {
+	my $h = $g->[ _V ]->_is_HYPER;
+	for my $v ( @_ ) {
+	    my @v = ref $v eq 'ARRAY' && $h ? @$v : $v;
+	    return () unless $V->has_path( @v );
+	    push @e, $V->_get_path_id( @v );
+	}
+    }
+    return @e;
+}
+
+sub has_edge {
+    my $g = shift;
+    my $E = $g->[ _E ];
+    my $V = $g->[ _V ];
+    my @i;
+    if (($V->[ _f ] & _LIGHT) && @_ == 2) {
+	return 0 unless
+	    exists $V->[ _s ]->{ $_[0] } &&
+	    exists $V->[ _s ]->{ $_[1] };
+	@i = @{ $V->[ _s ] }{ @_[ 0, 1 ] };
+    } else {
+	@i = $g->_vertex_ids( @_ );
+	return 0 if @i == 0 && @_;
+    }
+    my $f = $E->[ _f ];
+    if ($E->[ _a ] == 2 && @i == 2 && !($f & (_HYPER|_REF|_UNIQ))) { # Fast path.
+	@i = sort @i if ($f & _UNORD);
+	return exists $E->[ _s ]->{ $i[0] } &&
+	       exists $E->[ _s ]->{ $i[0] }->{ $i[1] } ? 1 : 0;
+    } else {
+	return defined $E->_get_path_id( @i ) ? 1 : 0;
+    }
+}
+
+sub edges05 {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    my @e = $g->[ _E ]->paths( @_ );
+    wantarray ?
+	map { [ map { my @v = $V->_get_id_path($_);
+		      @v == 1 ? $v[0] : [ @v ] }
+		@$_ ] }
+            @e : @e;
+}
+
+sub edges02 {
+    my $g = shift;
+    if (@_ && defined $_[0]) {
+	unless (defined $_[1]) {
+	    my @e = $g->edges_at($_[0]);
+	    wantarray ?
+		map { @$_ }
+                    sort { $a->[0] cmp $b->[0] || $a->[1] cmp $b->[1] } @e
+                : @e;
+	} else {
+	    die "edges02: unimplemented option";
+	}
+    } else {
+	my @e = map { ($_) x $g->get_edge_count(@$_) } $g->edges05( @_ );
+	wantarray ?
+          map { @$_ }
+              sort { $a->[0] cmp $b->[0] || $a->[1] cmp $b->[1] } @e
+          : @e;
+    }
+}
+
+sub unique_edges {
+    my $g = shift;
+    ($g->is_compat02) ? $g->edges02( @_ ) : $g->edges05( @_ );
+}
+
+sub edges {
+    my $g = shift;
+    if ($g->is_compat02) {
+	return $g->edges02( @_ );
+    } else {
+	if ($g->is_multiedged || $g->is_countedged) {
+	    if (wantarray) {
+		my @E;
+		for my $e ( $g->edges05 ) {
+		    push @E, ($e) x $g->get_edge_count(@$e);
+		}
+		return @E;
+	    } else {
+		my $E = 0;
+		for my $e ( $g->edges05 ) {
+		    $E += $g->get_edge_count(@$e);
+		}
+		return $E;
+	    }
+	} else {
+	    return $g->edges05;
+	}
+    }
+}
+
+sub has_edges {
+    my $g = shift;
+    scalar $g->[ _E ]->has_paths( @_ );
+}
+
+###
+# by_id
+#
+
+sub add_vertex_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    $g->[ _V ]->set_path_by_multi_id( @_ );
+    $g->[ _G ]++;
+    $g->_union_find_add_vertex( @_ ) if $g->has_union_find;
+    return $g;
+}
+
+sub add_vertex_get_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $id = $g->[ _V ]->set_path_by_multi_id( @_, _GEN_ID );
+    $g->[ _G ]++;
+    $g->_union_find_add_vertex( @_ ) if $g->has_union_find;
+    return $id;
+}
+
+sub has_vertex_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    $g->[ _V ]->has_path_by_multi_id( @_ );
+}
+
+sub delete_vertex_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $V = $g->[ _V ];
+    return unless $V->has_path_by_multi_id( @_ );
+    # TODO: what to about the edges at this vertex?
+    # If the multiness of this vertex goes to zero, delete the edges?
+    $V->del_path_by_multi_id( @_ );
+    $g->[ _G ]++;
+    return $g;
+}
+
+sub get_multivertex_ids {
+    my $g = shift;
+    $g->expect_multivertexed;
+    $g->[ _V ]->get_multi_ids( @_ );
+}
+
+sub add_edge_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $id = pop;
+    my @e = $g->_add_edge( @_ );
+    $g->[ _E ]->set_path( @e, $id );
+    $g->[ _G ]++;
+    $g->_union_find_add_edge( @e ) if $g->has_union_find;
+    return $g;
+}
+
+sub add_edge_get_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my @i = $g->_add_edge( @_ );
+    my $id = $g->[ _E ]->set_path_by_multi_id( @i, _GEN_ID );
+    $g->_union_find_add_edge( @i ) if $g->has_union_find;
+    $g->[ _G ]++;
+    return $id;
+}
+
+sub has_edge_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $id = pop;
+    my @i = $g->_vertex_ids( @_ );
+    return 0 if @i == 0 && @_;
+    $g->[ _E ]->has_path_by_multi_id( @i, $id );
+}
+
+sub delete_edge_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $V = $g->[ _E ];
+    my $id = pop;
+    my @i = $g->_vertex_ids( @_ );
+    return unless $V->has_path_by_multi_id( @i, $id );
+    $V->del_path_by_multi_id( @i, $id );
+    $g->[ _G ]++;
+    return $g;
+}
+
+sub get_multiedge_ids {
+    my $g = shift;
+    $g->expect_multiedged;
+    my @id = $g->_vertex_ids( @_ );
+    return unless @id;
+    $g->[ _E ]->get_multi_ids( @id );
+}
+
+###
+# Neighbourhood.
+#
+
+sub vertices_at {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    return @_ unless ($V->[ _f ] & _HYPER);
+    my %v;
+    my @i;
+    for my $v ( @_ ) {
+	my $i = $V->_get_path_id( $v );
+	return unless defined $i;
+	push @i, ( $v{ $v } = $i );
+    }
+    my $Vi = $V->_ids;
+    my @v;
+    while (my ($i, $v) = each %{ $Vi }) {
+	my %i;
+	my $h = $V->[_f ] & _HYPER;
+	@i{ @i } = @i if @i; # @todo: nonuniq hyper vertices?
+	for my $u (ref $v eq 'ARRAY' && $h ? @$v : $v) {
+	    my $j = exists $v{ $u } ? $v{ $u } : ( $v{ $u } = $i );
+	    if (defined $j && exists $i{ $j }) {
+		delete $i{ $j };
+		unless (keys %i) {
+		    push @v, $v;
+		    last;
+		}
+	    }
+	}
+    }
+    return @v;
+}
+
+sub _edges_at {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    my $E = $g->[ _E ];
+    my @e;
+    my $en = 0;
+    my %ev;
+    my $h = $V->[_f ] & _HYPER;
+    for my $v ( $h ? $g->vertices_at( @_ ) : @_ ) {
+	my $vi = $V->_get_path_id( ref $v eq 'ARRAY' && $h ? @$v : $v );
+	next unless defined $vi;
+	my $Ei = $E->_ids;
+	while (my ($ei, $ev) = each %{ $Ei }) {
+	    if (wantarray) {
+		for my $j (@$ev) {
+		    push @e, [ $ei, $ev ]
+			if $j == $vi && !$ev{$ei}++;
+		}
+	    } else {
+		for my $j (@$ev) {
+		    $en++ if $j == $vi;
+		}
+	    }		    
+	}
+    }
+    return wantarray ? @e : $en;
+}
+
+sub _edges_from {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    my $E = $g->[ _E ];
+    my @e;
+    my $o = $E->[ _f ] & _UNORD;
+    my $en = 0;
+    my %ev;
+    my $h = $V->[_f ] & _HYPER;
+    for my $v ( $h ? $g->vertices_at( @_ ) : @_ ) {
+	my $vi = $V->_get_path_id( ref $v eq 'ARRAY' && $h ? @$v : $v );
+	next unless defined $vi;
+	my $Ei = $E->_ids;
+	if (wantarray) {
+	    if ($o) {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    push @e, [ $ei, $ev ]
+			if ($ev->[0] == $vi || $ev->[-1] == $vi) && !$ev{$ei}++;
+		}
+	    } else {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    push @e, [ $ei, $ev ]
+			if $ev->[0] == $vi && !$ev{$ei}++;
+		}
+	    }
+	} else {
+	    if ($o) {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    $en++ if ($ev->[0] == $vi || $ev->[-1] == $vi);
+		}
+	    } else {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    $en++ if $ev->[0] == $vi;
+		}
+	    }
+	}
+    }
+    if (wantarray && $g->is_undirected) {
+	my @i = map { $V->_get_path_id( $_ ) } @_;
+	for my $e ( @e ) {
+	    unless ( $e->[ 1 ]->[ 0 ] == $i[ 0 ] ) { # @todo
+		$e = [ $e->[ 0 ], [ reverse @{ $e->[ 1 ] } ] ];
+	    }
+	}
+    }
+    return wantarray ? @e : $en;
+}
+
+sub _edges_to {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    my $E = $g->[ _E ];
+    my @e;
+    my $o = $E->[ _f ] & _UNORD;
+    my $en = 0;
+    my %ev;
+    my $h = $V->[_f ] & _HYPER;
+    for my $v ( $h ? $g->vertices_at( @_ ) : @_ ) {
+	my $vi = $V->_get_path_id( ref $v eq 'ARRAY' && $h ? @$v : $v );
+	next unless defined $vi;
+	my $Ei = $E->_ids;
+	if (wantarray) {
+	    if ($o) {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    push @e, [ $ei, $ev ]
+			if ($ev->[-1] == $vi || $ev->[0] == $vi) && !$ev{$ei}++;
+		}
+	    } else {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    push @e, [ $ei, $ev ]
+			if $ev->[-1] == $vi && !$ev{$ei}++;
+		}
+	    }
+	} else {
+	    if ($o) {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    $en++ if $ev->[-1] == $vi || $ev->[0] == $vi;
+		}
+	    } else {
+		while (my ($ei, $ev) = each %{ $Ei }) {
+		    next unless @$ev;
+		    $en++ if $ev->[-1] == $vi;
+		}
+	    }
+	}
+    }
+    if (wantarray && $g->is_undirected) {
+	my @i = map { $V->_get_path_id( $_ ) } @_;
+	for my $e ( @e ) {
+	    unless ( $e->[ 1 ]->[ -1 ] == $i[ -1 ] ) { # @todo
+		$e = [ $e->[ 0 ], [ reverse @{ $e->[ 1 ] } ] ];
+	    }
+	}
+    }
+    return wantarray ? @e : $en;
+}
+
+sub _edges_id_path {
+    my $g = shift;
+    my $V  = $g->[ _V ];
+    [ map { my @v = $V->_get_id_path($_);
+	    @v == 1 ? $v[0] : [ @v ] }
+          @{ $_[0]->[1] } ];
+}
+
+sub edges_at {
+    my $g = shift;
+    map { $g->_edges_id_path($_ ) } $g->_edges_at( @_ );
+}
+
+sub edges_from {
+    my $g = shift;
+    map { $g->_edges_id_path($_ ) } $g->_edges_from( @_ );
+}
+
+sub edges_to {
+    my $g = shift;
+    map { $g->_edges_id_path($_ ) } $g->_edges_to( @_ );
+}
+
+sub successors {
+    my $g = shift;
+    my $E = $g->[ _E ];
+    ($E->[ _f ] & _LIGHT) ?
+	$E->_successors($g, @_) :
+	Graph::AdjacencyMap::_successors($E, $g, @_);
+}
+
+sub predecessors {
+    my $g = shift;
+    my $E = $g->[ _E ];
+    ($E->[ _f ] & _LIGHT) ?
+	$E->_predecessors($g, @_) :
+	Graph::AdjacencyMap::_predecessors($E, $g, @_);
+}
+
+sub neighbours {
+    my $g = shift;
+    my $V  = $g->[ _V ];
+    my @s = map { my @v = @{ $_->[ 1 ] }; shift @v; @v } $g->_edges_from( @_ );
+    my @p = map { my @v = @{ $_->[ 1 ] }; pop   @v; @v } $g->_edges_to  ( @_ );
+    my %n;
+    @n{ @s } = @s;
+    @n{ @p } = @p;
+    map { $V->_get_id_path($_) } keys %n;
+}
+
+*neighbors = \&neighbours;
+
+sub delete_edge {
+    my $g = shift;
+    my @i = $g->_vertex_ids( @_ );
+    return $g unless @i;
+    my $i = $g->[ _E ]->_get_path_id( @i );
+    return $g unless defined $i;
+    $g->[ _E ]->_del_id( $i );
+    $g->[ _G ]++;
+    return $g;
+}
+
+sub delete_vertex {
+    my $g = shift;
+    my $V = $g->[ _V ];
+    return $g unless $V->has_path( @_ );
+    my $E = $g->[ _E ];
+    for my $e ( $g->_edges_at( @_ ) ) {
+	$E->_del_id( $e->[ 0 ] );
+    }
+    $V->del_path( @_ );
+    $g->[ _G ]++;
+    return $g;
+}
+
+sub get_vertex_count {
+    my $g = shift;
+    $g->[ _V ]->_get_path_count( @_ ) || 0;
+}
+
+sub get_edge_count {
+    my $g = shift;
+    my @e = $g->_vertex_ids( @_ );
+    return 0 unless @e;
+    $g->[ _E ]->_get_path_count( @e ) || 0;
+}
+
+sub delete_vertices {
+    my $g = shift;
+    while (@_) {
+	my $v = shift @_;
+	$g->delete_vertex($v);
+    }
+    return $g;
+}
+
+sub delete_edges {
+    my $g = shift;
+    while (@_) {
+	my ($u, $v) = splice @_, 0, 2;
+	$g->delete_edge($u, $v);
+    }
+    return $g;
+}
+
+###
+# Degrees.
+#
+
+sub _in_degree {
+    my $g = shift;
+    return undef unless @_ && $g->has_vertex( @_ );
+    my $in =  $g->is_undirected && $g->is_self_loop_vertex( @_ ) ? 1 : 0;
+    $in += $g->get_edge_count( @$_ ) for $g->edges_to( @_ );
+    return $in;
+}
+
+sub in_degree {
+    my $g = shift;
+    $g->_in_degree( @_ );
+}
+
+sub _out_degree {
+    my $g = shift;
+    return undef unless @_ && $g->has_vertex( @_ );
+    my $out =  $g->is_undirected && $g->is_self_loop_vertex( @_ ) ? 1 : 0;
+    $out += $g->get_edge_count( @$_ ) for $g->edges_from( @_ );
+    return $out;
+}
+
+sub out_degree {
+    my $g = shift;
+    $g->_out_degree( @_ );
+}
+
+sub _total_degree {
+    my $g = shift;
+    return undef unless @_ && $g->has_vertex( @_ );
+    $g->is_undirected ?
+	$g->_in_degree( @_ ) :
+	$g-> in_degree( @_ ) - $g-> out_degree( @_ );
+}
+
+sub degree {
+    my $g = shift;
+    if (@_) {
+	$g->_total_degree( @_ );
+    } else {
+	if ($g->is_undirected) {
+	    my $total = 0;
+	    $total += $g->_total_degree( $_ ) for $g->vertices05;
+	    return $total;
+	} else {
+	    return 0;
+	}
+    }
+}
+
+*vertex_degree = \&degree;
+
+sub is_sink_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->successors( @_ ) == 0 && $g->predecessors( @_ ) > 0;
+}
+
+sub is_source_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->predecessors( @_ ) == 0 && $g->successors( @_ ) > 0;
+}
+
+sub is_successorless_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->successors( @_ ) == 0;
+}
+
+sub is_predecessorless_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->predecessors( @_ ) == 0;
+}
+
+sub is_successorful_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->successors( @_ ) > 0;
+}
+
+sub is_predecessorful_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->predecessors( @_ ) > 0;
+}
+
+sub is_isolated_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->predecessors( @_ ) == 0 && $g->successors( @_ ) == 0;
+}
+
+sub is_interior_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    my $p = $g->predecessors( @_ );
+    my $s = $g->successors( @_ );
+    if ($g->is_self_loop_vertex( @_ )) {
+	$p--;
+	$s--;
+    }
+    $p > 0 && $s > 0;
+}
+
+sub is_exterior_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    $g->predecessors( @_ ) == 0 || $g->successors( @_ ) == 0;
+}
+
+sub is_self_loop_vertex {
+    my $g = shift;
+    return 0 unless @_;
+    for my $s ( $g->successors( @_ ) ) {
+	return 1 if $s eq $_[0]; # @todo: hypervertices
+    }
+    return 0;
+}
+
+sub sink_vertices {
+    my $g = shift;
+    grep { $g->is_sink_vertex($_) } $g->vertices05;
+}
+
+sub source_vertices {
+    my $g = shift;
+    grep { $g->is_source_vertex($_) } $g->vertices05;
+}
+
+sub successorless_vertices {
+    my $g = shift;
+    grep { $g->is_successorless_vertex($_) } $g->vertices05;
+}
+
+sub predecessorless_vertices {
+    my $g = shift;
+    grep { $g->is_predecessorless_vertex($_) } $g->vertices05;
+}
+
+sub successorful_vertices {
+    my $g = shift;
+    grep { $g->is_successorful_vertex($_) } $g->vertices05;
+}
+
+sub predecessorful_vertices {
+    my $g = shift;
+    grep { $g->is_predecessorful_vertex($_) } $g->vertices05;
+}
+
+sub isolated_vertices {
+    my $g = shift;
+    grep { $g->is_isolated_vertex($_) } $g->vertices05;
+}
+
+sub interior_vertices {
+    my $g = shift;
+    grep { $g->is_interior_vertex($_) } $g->vertices05;
+}
+
+sub exterior_vertices {
+    my $g = shift;
+    grep { $g->is_exterior_vertex($_) } $g->vertices05;
+}
+
+sub self_loop_vertices {
+    my $g = shift;
+    grep { $g->is_self_loop_vertex($_) } $g->vertices05;
+}
+
+###
+# Paths and cycles.
+#
+
+sub add_path {
+    my $g = shift;
+    my $u = shift;
+    while (@_) {
+	my $v = shift;
+	$g->add_edge($u, $v);
+	$u = $v;
+    }
+    return $g;
+}
+
+sub delete_path {
+    my $g = shift;
+    my $u = shift;
+    while (@_) {
+	my $v = shift;
+	$g->delete_edge($u, $v);
+	$u = $v;
+    }
+    return $g;
+}
+
+sub has_path {
+    my $g = shift;
+    my $u = shift;
+    while (@_) {
+	my $v = shift;
+	return 0 unless $g->has_edge($u, $v);
+	$u = $v;
+    }
+    return $g;
+}
+
+sub add_cycle {
+    my $g = shift;
+    $g->add_path(@_, $_[0]);
+}
+
+sub delete_cycle {
+    my $g = shift;
+    $g->delete_path(@_, $_[0]);
+}
+
+sub has_cycle {
+    my $g = shift;
+    @_ ? ($g->has_path(@_, $_[0]) ? 1 : 0) : 0;
+}
+
+sub has_a_cycle {
+    my $g = shift;
+    my @r = ( back_edge => \&Graph::Traversal::has_a_cycle );
+    push @r,
+      down_edge => \&Graph::Traversal::has_a_cycle
+       if $g->is_undirected;
+    my $t = Graph::Traversal::DFS->new($g, @r, @_);
+    $t->dfs;
+    return $t->get_state('has_a_cycle');
+}
+
+sub find_a_cycle {
+    my $g = shift;
+    my @r = ( back_edge => \&Graph::Traversal::find_a_cycle);
+    push @r,
+      down_edge => \&Graph::Traversal::find_a_cycle
+	if $g->is_undirected;
+    my $t = Graph::Traversal::DFS->new($g, @r, @_);
+    $t->dfs;
+    $t->has_state('a_cycle') ? @{ $t->get_state('a_cycle') } : ();
+}
+
+###
+# Attributes.
+
+# Vertex attributes.
+
+sub set_vertex_attribute {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $value = pop;
+    my $attr  = pop;
+    $g->add_vertex( @_ ) unless $g->has_vertex( @_ );
+    $g->[ _V ]->_set_path_attr( @_, $attr, $value );
+}
+
+sub set_vertex_attribute_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $value = pop;
+    my $attr  = pop;
+    $g->add_vertex_by_id( @_ ) unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_set_path_attr( @_, $attr, $value );
+}
+
+sub set_vertex_attributes {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $attr = pop;
+    $g->add_vertex( @_ ) unless $g->has_vertex( @_ );
+    $g->[ _V ]->_set_path_attrs( @_, $attr );
+}
+
+sub set_vertex_attributes_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $attr = pop;
+    $g->add_vertex_by_id( @_ ) unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_set_path_attrs( @_, $attr );
+}
+
+sub has_vertex_attributes {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    return 0 unless $g->has_vertex( @_ );
+    $g->[ _V ]->_has_path_attrs( @_ );
+}
+
+sub has_vertex_attributes_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    return 0 unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_has_path_attrs( @_ );
+}
+
+sub has_vertex_attribute {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $attr = pop;
+    return 0 unless $g->has_vertex( @_ );
+    $g->[ _V ]->_has_path_attr( @_, $attr );
+}
+
+sub has_vertex_attribute_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $attr = pop;
+    return 0 unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_has_path_attr( @_, $attr );
+}
+
+sub get_vertex_attributes {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    return unless $g->has_vertex( @_ );
+    my $a = $g->[ _V ]->_get_path_attrs( @_ );
+    ($g->is_compat02) ? (defined $a ? %{ $a } : ()) : $a;
+}
+
+sub get_vertex_attributes_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    return unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_get_path_attrs( @_ );
+}
+
+sub get_vertex_attribute {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $attr = pop;
+    return unless $g->has_vertex( @_ );
+    $g->[ _V ]->_get_path_attr( @_, $attr );
+}
+
+sub get_vertex_attribute_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $attr = pop;
+    return unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_get_path_attr( @_, $attr );
+}
+
+sub get_vertex_attribute_names {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    return unless $g->has_vertex( @_ );
+    $g->[ _V ]->_get_path_attr_names( @_ );
+}
+
+sub get_vertex_attribute_names_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    return unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_get_path_attr_names( @_ );
+}
+
+sub get_vertex_attribute_values {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    return unless $g->has_vertex( @_ );
+    $g->[ _V ]->_get_path_attr_values( @_ );
+}
+
+sub get_vertex_attribute_values_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    return unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_get_path_attr_values( @_ );
+}
+
+sub delete_vertex_attributes {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    return undef unless $g->has_vertex( @_ );
+    $g->[ _V ]->_del_path_attrs( @_ );
+}
+
+sub delete_vertex_attributes_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    return undef unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_del_path_attrs( @_ );
+}
+
+sub delete_vertex_attribute {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $attr = pop;
+    return undef unless $g->has_vertex( @_ );
+    $g->[ _V ]->_del_path_attr( @_, $attr );
+}
+
+sub delete_vertex_attribute_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $attr = pop;
+    return undef unless $g->has_vertex_by_id( @_ );
+    $g->[ _V ]->_del_path_attr( @_, $attr );
+}
+
+# Edge attributes.
+
+sub _set_edge_attribute {
+    my $g = shift;
+    my $value = pop;
+    my $attr  = pop;
+    my $E = $g->[ _E ];
+    my $f = $E->[ _f ];
+    my @i;
+    if ($E->[ _a ] == 2 && @_ == 2 && !($f & (_HYPER|_REF|_UNIQ))) { # Fast path.
+	@_ = sort @_ if ($f & _UNORD);
+	my $s = $E->[ _s ];
+	$g->add_edge( @_ ) unless exists $s->{ $_[0] } && exists $s->{ $_[0] }->{ $_[1] };
+	@i = @{ $g->[ _V ]->[ _s ] }{ @_ };
+    } else {
+	$g->add_edge( @_ ) unless $g->has_edge( @_ );
+	@i = $g->_vertex_ids( @_ );
+    }
+    $g->[ _E ]->_set_path_attr( @i, $attr, $value );
+}
+
+sub set_edge_attribute {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $value = pop;
+    my $attr  = pop;
+    my $E = $g->[ _E ];
+    $g->add_edge( @_ ) unless $g->has_edge( @_ );
+    $E->_set_path_attr( $g->_vertex_ids( @_ ), $attr, $value );
+}
+
+sub set_edge_attribute_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $value = pop;
+    my $attr  = pop;
+    # $g->add_edge_by_id( @_ ) unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_set_path_attr( $g->_vertex_ids( @_ ), $id, $attr, $value );
+}
+
+sub set_edge_attributes {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $attr = pop;
+    $g->add_edge( @_ ) unless $g->has_edge( @_ );
+    $g->[ _E ]->_set_path_attrs( $g->_vertex_ids( @_ ), $attr );
+}
+
+sub set_edge_attributes_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $attr = pop;
+    $g->add_edge_by_id( @_ ) unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_set_path_attrs( $g->_vertex_ids( @_ ), $id, $attr );
+}
+
+sub has_edge_attributes {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    return 0 unless $g->has_edge( @_ );
+    $g->[ _E ]->_has_path_attrs( $g->_vertex_ids( @_ ) );
+}
+
+sub has_edge_attributes_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    return 0 unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_has_path_attrs( $g->_vertex_ids( @_ ), $id );
+}
+
+sub has_edge_attribute {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $attr = pop;
+    return 0 unless $g->has_edge( @_ );
+    $g->[ _E ]->_has_path_attr( $g->_vertex_ids( @_ ), $attr );
+}
+
+sub has_edge_attribute_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $attr = pop;
+    return 0 unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_has_path_attr( $g->_vertex_ids( @_ ), $id, $attr );
+}
+
+sub get_edge_attributes {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    return unless $g->has_edge( @_ );
+    my $a = $g->[ _E ]->_get_path_attrs( $g->_vertex_ids( @_ ) );
+    ($g->is_compat02) ? (defined $a ? %{ $a } : ()) : $a;
+}
+
+sub get_edge_attributes_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    return unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    return $g->[ _E ]->_get_path_attrs( $g->_vertex_ids( @_ ), $id );
+}
+
+sub _get_edge_attribute { # Fast path; less checks.
+    my $g = shift;
+    my $attr = pop;
+    my $E = $g->[ _E ];
+    my $f = $E->[ _f ];
+    if ($E->[ _a ] == 2 && @_ == 2 && !($f & (_HYPER|_REF|_UNIQ))) { # Fast path.
+	@_ = sort @_ if ($f & _UNORD);
+	my $s = $E->[ _s ];
+	return unless exists $s->{ $_[0] } && exists $s->{ $_[0] }->{ $_[1] };
+    } else {
+	return unless $g->has_edge( @_ );
+    }
+    my @i = $g->_vertex_ids( @_ );
+    $E->_get_path_attr( @i, $attr );
+}
+
+sub get_edge_attribute {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $attr = pop;
+    return undef unless $g->has_edge( @_ );
+    my @i = $g->_vertex_ids( @_ );
+    return undef if @i == 0 && @_;
+    my $E = $g->[ _E ];
+    $E->_get_path_attr( @i, $attr );
+}
+
+sub get_edge_attribute_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $attr = pop;
+    return unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_get_path_attr( $g->_vertex_ids( @_ ), $id, $attr );
+}
+
+sub get_edge_attribute_names {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    return unless $g->has_edge( @_ );
+    $g->[ _E ]->_get_path_attr_names( $g->_vertex_ids( @_ ) );
+}
+
+sub get_edge_attribute_names_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    return unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_get_path_attr_names( $g->_vertex_ids( @_ ), $id );
+}
+
+sub get_edge_attribute_values {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    return unless $g->has_edge( @_ );
+    $g->[ _E ]->_get_path_attr_values( $g->_vertex_ids( @_ ) );
+}
+
+sub get_edge_attribute_values_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    return unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_get_path_attr_values( $g->_vertex_ids( @_ ), $id );
+}
+
+sub delete_edge_attributes {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    return unless $g->has_edge( @_ );
+    $g->[ _E ]->_del_path_attrs( $g->_vertex_ids( @_ ) );
+}
+
+sub delete_edge_attributes_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    return unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_del_path_attrs( $g->_vertex_ids( @_ ), $id );
+}
+
+sub delete_edge_attribute {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $attr = pop;
+    return unless $g->has_edge( @_ );
+    $g->[ _E ]->_del_path_attr( $g->_vertex_ids( @_ ), $attr );
+}
+
+sub delete_edge_attribute_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $attr = pop;
+    return unless $g->has_edge_by_id( @_ );
+    my $id = pop;
+    $g->[ _E ]->_del_path_attr( $g->_vertex_ids( @_ ), $id, $attr );
+}
+
+###
+# Compat.
+#
+
+sub vertex {
+    my $g = shift;
+    $g->has_vertex( @_ ) ? @_ : undef;
+}
+
+sub out_edges {
+    my $g = shift;
+    return unless @_ && $g->has_vertex( @_ );
+    my @e = $g->edges_from( @_ );
+    wantarray ? map { @$_ } @e : @e;
+}
+
+sub in_edges {
+    my $g = shift;
+    return unless @_ && $g->has_vertex( @_ );
+    my @e = $g->edges_to( @_ );
+    wantarray ? map { @$_ } @e : @e;
+}
+
+sub add_vertices {
+    my $g = shift;
+    $g->add_vertex( $_ ) for @_;
+}
+
+sub add_edges {
+    my $g = shift;
+    while (@_) {
+	my $u = shift @_;
+	if (ref $u eq 'ARRAY') {
+	    $g->add_edge( @$u );
+	} else {
+	    if (@_) {
+		my $v = shift @_;
+		$g->add_edge( $u, $v );
+	    } else {
+		require Carp;
+		Carp::croak("Graph::add_edges: missing end vertex");
+	    }
+	}
+    }
+}
+
+###
+# More constructors.
+#
+
+sub copy {
+    my $g = shift;
+    my %opt = _get_options( \@_ );
+    
+    my $c = (ref $g)->new(directed => $g->directed ? 1 : 0,
+			  compat02 => $g->compat02 ? 1 : 0);
+    for my $v ($g->isolated_vertices) { $c->add_vertex($v) }
+    for my $e ($g->edges05)           { $c->add_edge(@$e)  }
+    return $c;
+}
+
+*copy_graph = \©
+
+sub deep_copy {
+    require Data::Dumper;
+    my $g = shift;
+    my $d = Data::Dumper->new([$g]);
+    use vars qw($VAR1);
+    $d->Purity(1)->Terse(1)->Deepcopy(1);
+    $d->Deparse(1) if $] >= 5.008;
+    eval $d->Dump;
+}
+
+*deep_copy_graph = \&deep_copy;
+
+sub transpose_edge {
+    my $g = shift;
+    if ($g->is_directed) {
+	return undef unless $g->has_edge( @_ );
+	my $c = $g->get_edge_count( @_ );
+	my $a = $g->get_edge_attributes( @_ );
+	my @e = reverse @_;
+	$g->delete_edge( @_ ) unless $g->has_edge( @e );
+	$g->add_edge( @e ) for 1..$c;
+	$g->set_edge_attributes(@e, $a) if $a;
+    }
+    return $g;
+}
+
+sub transpose_graph {
+    my $g = shift;
+    my $t = $g->copy;
+    if ($t->directed) {
+	for my $e ($t->edges05) {
+	    $t->transpose_edge(@$e);
+	}
+    }
+    return $t;
+}
+
+*transpose = \&transpose_graph;
+
+sub complete_graph {
+    my $g = shift;
+    my $c = $g->new( directed => $g->directed );
+    my @v = $g->vertices05;
+    for (my $i = 0; $i <= $#v; $i++ ) {
+	for (my $j = 0; $j <= $#v; $j++ ) {
+	    next if $i >= $j;
+	    if ($g->is_undirected) {
+		$c->add_edge($v[$i], $v[$j]);
+	    } else {
+		$c->add_edge($v[$i], $v[$j]);
+		$c->add_edge($v[$j], $v[$i]);
+	    }
+	}
+    }
+    return $c;
+}
+
+*complement = \&complement_graph;
+
+sub complement_graph {
+    my $g = shift;
+    my $c = $g->new( directed => $g->directed );
+    my @v = $g->vertices05;
+    for (my $i = 0; $i <= $#v; $i++ ) {
+	for (my $j = 0; $j <= $#v; $j++ ) {
+	    next if $i >= $j;
+	    if ($g->is_undirected) {
+		$c->add_edge($v[$i], $v[$j])
+		    unless $g->has_edge($v[$i], $v[$j]);
+	    } else {
+		$c->add_edge($v[$i], $v[$j])
+		    unless $g->has_edge($v[$i], $v[$j]);
+		$c->add_edge($v[$j], $v[$i])
+		    unless $g->has_edge($v[$j], $v[$i]);
+	    }
+	}
+    }
+    return $c;
+}
+
+*complete = \&complete_graph;
+
+###
+# Transitivity.
+#
+
+sub is_transitive {
+    my $g = shift;
+    Graph::TransitiveClosure::is_transitive($g);
+}
+
+###
+# Weighted vertices.
+#
+
+my $defattr = 'weight';
+
+sub _defattr {
+    return $defattr;
+}
+
+sub add_weighted_vertex {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $w = pop;
+    $g->add_vertex(@_);
+    $g->set_vertex_attribute(@_, $defattr, $w);
+}
+
+sub add_weighted_vertices {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    while (@_) {
+	my ($v, $w) = splice @_, 0, 2;
+	$g->add_vertex($v);
+	$g->set_vertex_attribute($v, $defattr, $w);
+    }
+}
+
+sub get_vertex_weight {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    $g->get_vertex_attribute(@_, $defattr);
+}
+
+sub has_vertex_weight {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    $g->has_vertex_attribute(@_, $defattr);
+}
+
+sub set_vertex_weight {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    my $w = pop;
+    $g->set_vertex_attribute(@_, $defattr, $w);
+}
+
+sub delete_vertex_weight {
+    my $g = shift;
+    $g->expect_non_multivertexed;
+    $g->delete_vertex_attribute(@_, $defattr);
+}
+
+sub add_weighted_vertex_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $w = pop;
+    $g->add_vertex_by_id(@_);
+    $g->set_vertex_attribute_by_id(@_, $defattr, $w);
+}
+
+sub add_weighted_vertices_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $id = pop;
+    while (@_) {
+	my ($v, $w) = splice @_, 0, 2;
+	$g->add_vertex_by_id($v, $id);
+	$g->set_vertex_attribute_by_id($v, $id, $defattr, $w);
+    }
+}
+
+sub get_vertex_weight_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    $g->get_vertex_attribute_by_id(@_, $defattr);
+}
+
+sub has_vertex_weight_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    $g->has_vertex_attribute_by_id(@_, $defattr);
+}
+
+sub set_vertex_weight_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    my $w = pop;
+    $g->set_vertex_attribute_by_id(@_, $defattr, $w);
+}
+
+sub delete_vertex_weight_by_id {
+    my $g = shift;
+    $g->expect_multivertexed;
+    $g->delete_vertex_attribute_by_id(@_, $defattr);
+}
+
+###
+# Weighted edges.
+#
+
+sub add_weighted_edge {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    if ($g->is_compat02) {
+	my $w = splice @_, 1, 1;
+	$g->add_edge(@_);
+	$g->set_edge_attribute(@_, $defattr, $w);
+    } else {
+	my $w = pop;
+	$g->add_edge(@_);
+	$g->set_edge_attribute(@_, $defattr, $w);
+    }
+}
+
+sub add_weighted_edges {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    if ($g->is_compat02) {
+	while (@_) {
+	    my ($u, $w, $v) = splice @_, 0, 3;
+	    $g->add_edge($u, $v);
+	    $g->set_edge_attribute($u, $v, $defattr, $w);
+	}
+    } else {
+	while (@_) {
+	    my ($u, $v, $w) = splice @_, 0, 3;
+	    $g->add_edge($u, $v);
+	    $g->set_edge_attribute($u, $v, $defattr, $w);
+	}
+    }
+}
+
+sub add_weighted_edges_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $id = pop;
+    while (@_) {
+	my ($u, $v, $w) = splice @_, 0, 3;
+	$g->add_edge_by_id($u, $v, $id);
+	$g->set_edge_attribute_by_id($u, $v, $id, $defattr, $w);
+    }
+}
+
+sub add_weighted_path {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $u = shift;
+    while (@_) {
+	my ($w, $v) = splice @_, 0, 2;
+	$g->add_edge($u, $v);
+	$g->set_edge_attribute($u, $v, $defattr, $w);
+	$u = $v;
+    }
+}
+
+sub get_edge_weight {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    $g->get_edge_attribute(@_, $defattr);
+}
+
+sub has_edge_weight {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    $g->has_edge_attribute(@_, $defattr);
+}
+
+sub set_edge_weight {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    my $w = pop;
+    $g->set_edge_attribute(@_, $defattr, $w);
+}
+
+sub delete_edge_weight {
+    my $g = shift;
+    $g->expect_non_multiedged;
+    $g->delete_edge_attribute(@_, $defattr);
+}
+
+sub add_weighted_edge_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    if ($g->is_compat02) {
+	my $w = splice @_, 1, 1;
+	$g->add_edge_by_id(@_);
+	$g->set_edge_attribute_by_id(@_, $defattr, $w);
+    } else {
+	my $w = pop;
+	$g->add_edge_by_id(@_);
+	$g->set_edge_attribute_by_id(@_, $defattr, $w);
+    }
+}
+
+sub add_weighted_path_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $id = pop;
+    my $u = shift;
+    while (@_) {
+	my ($w, $v) = splice @_, 0, 2;
+	$g->add_edge_by_id($u, $v, $id);
+	$g->set_edge_attribute_by_id($u, $v, $id, $defattr, $w);
+	$u = $v;
+    }
+}
+
+sub get_edge_weight_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    $g->get_edge_attribute_by_id(@_, $defattr);
+}
+
+sub has_edge_weight_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    $g->has_edge_attribute_by_id(@_, $defattr);
+}
+
+sub set_edge_weight_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    my $w = pop;
+    $g->set_edge_attribute_by_id(@_, $defattr, $w);
+}
+
+sub delete_edge_weight_by_id {
+    my $g = shift;
+    $g->expect_multiedged;
+    $g->delete_edge_attribute_by_id(@_, $defattr);
+}
+
+###
+# Error helpers.
+#
+
+my %expected;
+@expected{qw(directed undirected acyclic)} = qw(undirected directed cyclic);
+
+sub _expected {
+    my $exp = shift;
+    my $got = @_ ? shift : $expected{$exp};
+    $got = defined $got ? ", got $got" : "";
+    if (my @caller2 = caller(2)) {
+	die "$caller2[3]: expected $exp graph$got, at $caller2[1] line $caller2[2].\n";
+    } else {
+	my @caller1 = caller(1);
+	die "$caller1[3]: expected $exp graph$got, at $caller1[1] line $caller1[2].\n";
+    }
+}
+
+sub expect_undirected {
+    my $g = shift;
+    _expected('undirected') unless $g->is_undirected;
+}
+
+sub expect_directed {
+    my $g = shift;
+    _expected('directed') unless $g->is_directed;
+}
+
+sub expect_acyclic {
+    my $g = shift;
+    _expected('acyclic') unless $g->is_acyclic;
+}
+
+sub expect_dag {
+    my $g = shift;
+    my @got;
+    push @got, 'undirected' unless $g->is_directed;
+    push @got, 'cyclic'     unless $g->is_acyclic;
+    _expected('directed acyclic', "@got") if @got;
+}
+
+sub expect_multivertexed {
+    my $g = shift;
+    _expected('multivertexed') unless $g->is_multivertexed;
+}
+
+sub expect_non_multivertexed {
+    my $g = shift;
+    _expected('non-multivertexed') if $g->is_multivertexed;
+}
+
+sub expect_non_multiedged {
+    my $g = shift;
+    _expected('non-multiedged') if $g->is_multiedged;
+}
+
+sub expect_multiedged {
+    my $g = shift;
+    _expected('multiedged') unless $g->is_multiedged;
+}
+
+sub _get_options {
+    my @caller = caller(1);
+    unless (@_ == 1 && ref $_[0] eq 'ARRAY') {
+	die "$caller[3]: internal error: should be called with only one array ref argument, at $caller[1] line $caller[2].\n";
+    }
+    my @opt = @{ $_[0] };
+    unless (@opt  % 2 == 0) {
+	die "$caller[3]: expected an options hash, got a non-even number of arguments, at $caller[1] line $caller[2].\n";
+    }
+    return @opt;
+}
+
+###
+# Random constructors and accessors.
+#
+
+sub __fisher_yates_shuffle (@) {
+    # From perlfaq4, but modified to be non-modifying.
+    my @a = @_;
+    my $i = @a;
+    while ($i--) {
+	my $j = int rand ($i+1);
+	@a[$i,$j] = @a[$j,$i];
+    }
+    return @a;
+}
+
+BEGIN {
+    sub _shuffle(@);
+    # Workaround for the Perl bug [perl #32383] where -d:Dprof and
+    # List::Util::shuffle do not like each other: if any debugging
+    # (-d) flags are on, fall back to our own Fisher-Yates shuffle.
+    # The bug was fixed by perl changes #26054 and #26062, which
+    # went to Perl 5.9.3.  If someone tests this with a pre-5.9.3
+    # bleadperl that calls itself 5.9.3 but doesn't yet have the
+    # patches, oh, well.
+    *_shuffle = $^P && $] < 5.009003 ?
+	\&__fisher_yates_shuffle : \&List::Util::shuffle;
+}
+
+sub random_graph {
+    my $class = (@_ % 2) == 0 ? 'Graph' : shift;
+    my %opt = _get_options( \@_ );
+    my $random_edge;
+    unless (exists $opt{vertices} && defined $opt{vertices}) {
+	require Carp;
+	Carp::croak("Graph::random_graph: argument 'vertices' missing or undef");
+    }
+    if (exists $opt{random_seed}) {
+	srand($opt{random_seed});
+	delete $opt{random_seed};
+    }
+    if (exists $opt{random_edge}) {
+	$random_edge = $opt{random_edge};
+	delete $opt{random_edge};
+    }
+    my @V;
+    if (my $ref = ref $opt{vertices}) {
+	if ($ref eq 'ARRAY') {
+	    @V = @{ $opt{vertices} };
+	} else {
+	    Carp::croak("Graph::random_graph: argument 'vertices' illegal");
+	}
+    } else {
+	@V = 0..($opt{vertices} - 1);
+    }
+    delete $opt{vertices};
+    my $V = @V;
+    my $C = $V * ($V - 1) / 2;
+    my $E;
+    if (exists $opt{edges} && exists $opt{edges_fill}) {
+	Carp::croak("Graph::random_graph: both arguments 'edges' and 'edges_fill' specified");
+    }
+    $E = exists $opt{edges_fill} ? $opt{edges_fill} * $C : $opt{edges};
+    delete $opt{edges};
+    delete $opt{edges_fill};
+    my $g = $class->new(%opt);
+    $g->add_vertices(@V);
+    return $g if $V < 2;
+    $C *= 2 if $g->directed;
+    $E = $C / 2 unless defined $E;
+    $E = int($E + 0.5);
+    my $p = $E / $C;
+    $random_edge = sub { $p } unless defined $random_edge;
+    # print "V = $V, E = $E, C = $C, p = $p\n";
+    if ($p > 1.0 && !($g->countedged || $g->multiedged)) {
+	require Carp;
+	Carp::croak("Graph::random_graph: needs to be countedged or multiedged ($E > $C)");
+    }
+    my @V1 = @V;
+    my @V2 = @V;
+    # Shuffle the vertex lists so that the pairs at
+    # the beginning of the lists are not more likely.
+    @V1 = _shuffle @V1;
+    @V2 = _shuffle @V2;
+ LOOP:
+    while ($E) {
+	for my $v1 (@V1) {
+	    for my $v2 (@V2) {
+		next if $v1 eq $v2; # TODO: allow self-loops?
+		my $q = $random_edge->($g, $v1, $v2, $p);
+		if ($q && ($q == 1 || rand() <= $q) &&
+		    !$g->has_edge($v1, $v2)) {
+		    $g->add_edge($v1, $v2);
+		    $E--;
+		    last LOOP unless $E;
+		}
+	    }
+	}
+    }
+    return $g;
+}
+
+sub random_vertex {
+    my $g = shift;
+    my @V = $g->vertices05;
+    @V[rand @V];
+}
+
+sub random_edge {
+    my $g = shift;
+    my @E = $g->edges05;
+    @E[rand @E];
+}
+
+sub random_successor {
+    my ($g, $v) = @_;
+    my @S = $g->successors($v);
+    @S[rand @S];
+}
+
+sub random_predecessor {
+    my ($g, $v) = @_;
+    my @P = $g->predecessors($v);
+    @P[rand @P];
+}
+
+###
+# Algorithms.
+#
+
+my $MST_comparator = sub { ($_[0] || 0) <=> ($_[1] || 0) };
+
+sub _MST_attr {
+    my $attr = shift;
+    my $attribute =
+	exists $attr->{attribute}  ?
+	    $attr->{attribute}  : $defattr;
+    my $comparator =
+	exists $attr->{comparator} ?
+	    $attr->{comparator} : $MST_comparator;
+    return ($attribute, $comparator);
+}
+
+sub _MST_edges {
+    my ($g, $attr) = @_;
+    my ($attribute, $comparator) = _MST_attr($attr);
+    map { $_->[1] }
+        sort { $comparator->($a->[0], $b->[0], $a->[1], $b->[1]) }
+             map { [ $g->get_edge_attribute(@$_, $attribute), $_ ] }
+                 $g->edges05;
+}
+
+sub MST_Kruskal {
+    my ($g, %attr) = @_;
+
+    $g->expect_undirected;
+
+    my $MST = Graph::Undirected->new;
+
+    my $UF  = Graph::UnionFind->new;
+    for my $v ($g->vertices05) { $UF->add($v) }
+
+    for my $e ($g->_MST_edges(\%attr)) {
+	my ($u, $v) = @$e; # TODO: hyperedges
+	my $t0 = $UF->find( $u );
+	my $t1 = $UF->find( $v );
+	unless ($t0 eq $t1) {
+	    $UF->union($u, $v);
+	    $MST->add_edge($u, $v);
+	}
+    }
+
+    return $MST;
+}
+
+sub _MST_add {
+    my ($g, $h, $HF, $r, $attr, $unseen) = @_;
+    for my $s ( grep { exists $unseen->{ $_ } } $g->successors( $r ) ) {
+	$HF->add( Graph::MSTHeapElem->new( $r, $s, $g->get_edge_attribute( $r, $s, $attr ) ) );
+    }
+}
+
+sub _next_alphabetic { shift; (sort               keys %{ $_[0] })[0] }
+sub _next_numeric    { shift; (sort { $a <=> $b } keys %{ $_[0] })[0] }
+sub _next_random     { shift; (values %{ $_[0] })[ rand keys %{ $_[0] } ] }
+
+sub _root_opt {
+    my $g = shift;
+    my %opt = @_ == 1 ? ( first_root => $_[0] ) : _get_options( \@_ );
+    my %unseen;
+    my @unseen = $g->vertices05;
+    @unseen{ @unseen } = @unseen;
+    @unseen = _shuffle @unseen;
+    my $r;
+    if (exists $opt{ start }) {
+	$opt{ first_root } = $opt{ start };
+	$opt{ next_root  } = undef;
+    }
+    if (exists $opt{ get_next_root }) {
+	$opt{ next_root  } = $opt{ get_next_root }; # Graph 0.201 compat.
+    }
+    if (exists $opt{ first_root }) {
+	if (ref $opt{ first_root } eq 'CODE') {
+	    $r = $opt{ first_root }->( $g, \%unseen );
+	} else {
+	    $r = $opt{ first_root };
+	}
+    } else {
+	$r = shift @unseen;
+    }
+    my $next =
+	exists $opt{ next_root } ?
+	    $opt{ next_root } :
+		$opt{ next_alphabetic } ?
+		    \&_next_alphabetic :
+			$opt{ next_numeric } ? \&_next_numeric :
+			    \&_next_random;
+    my $code = ref $next eq 'CODE';
+    my $attr = exists $opt{ attribute } ? $opt{ attribute } : $defattr;
+    return ( \%opt, \%unseen, \@unseen, $r, $next, $code, $attr );
+}
+
+sub _heap_walk {
+    my ($g, $h, $add, $etc) = splice @_, 0, 4; # Leave %opt in @_.
+
+    my ($opt, $unseenh, $unseena, $r, $next, $code, $attr) = $g->_root_opt(@_);
+    my $HF = Heap071::Fibonacci->new;
+
+    while (defined $r) {
+	# print "r = $r\n";
+	$add->($g, $h, $HF, $r, $attr, $unseenh, $etc);
+	delete $unseenh->{ $r };
+	while (defined $HF->top) {
+	    my $t = $HF->extract_top;
+	    # use Data::Dumper; print "t = ", Dumper($t);
+	    if (defined $t) {
+		my ($u, $v, $w) = $t->val;
+		# print "extracted top: $u $v $w\n";
+		if (exists $unseenh->{ $v }) {
+		    $h->set_edge_attribute($u, $v, $attr, $w);
+		    delete $unseenh->{ $v };
+		    $add->($g, $h, $HF, $v, $attr, $unseenh, $etc);
+		}
+	    }
+	}
+	return $h unless defined $next;
+	$r = $code ? $next->( $g, $unseenh ) : shift @$unseena;
+    }
+
+    return $h;
+}
+
+sub MST_Prim {
+    my $g = shift;
+    $g->expect_undirected;
+    $g->_heap_walk(Graph::Undirected->new(), \&_MST_add, undef, @_);
+}
+
+*MST_Dijkstra = \&MST_Prim;
+
+*minimum_spanning_tree = \&MST_Prim;
+
+###
+# Cycle detection.
+#
+
+*is_cyclic = \&has_a_cycle;
+
+sub is_acyclic {
+    my $g = shift;
+    return !$g->is_cyclic;
+}
+
+sub is_dag {
+    my $g = shift;
+    return $g->is_directed && $g->is_acyclic ? 1 : 0;
+}
+
+*is_directed_acyclic_graph = \&is_dag;
+
+###
+# Backward compat.
+#
+
+sub average_degree {
+    my $g = shift;
+    my $V = $g->vertices05;
+
+    return $V ? $g->degree / $V : 0;
+}
+
+sub density_limits {
+    my $g = shift;
+
+    my $V = $g->vertices05;
+    my $M = $V * ($V - 1);
+
+    $M /= 2 if $g->is_undirected;
+
+    return ( 0.25 * $M, 0.75 * $M, $M );
+}
+
+sub density {
+    my $g = shift;
+    my ($sparse, $dense, $complete) = $g->density_limits;
+
+    return $complete ? $g->edges / $complete : 0;
+}
+
+###
+# Attribute backward compat
+#
+
+sub _attr02_012 {
+    my ($g, $op, $ga, $va, $ea) = splice @_, 0, 5;
+    if ($g->is_compat02) {
+	if    (@_ == 0) { return $ga->( $g ) }
+	elsif (@_ == 1) { return $va->( $g, @_ ) }
+	elsif (@_ == 2) { return $ea->( $g, @_ ) }
+	else {
+	    die sprintf "$op: wrong number of arguments (%d)", scalar @_;
+	}
+    } else {
+	die "$op: not a compat02 graph"
+    }
+}
+
+sub _attr02_123 {
+    my ($g, $op, $ga, $va, $ea) = splice @_, 0, 5;
+    if ($g->is_compat02) {
+	if    (@_ == 1) { return $ga->( $g, @_ ) }
+	elsif (@_ == 2) { return $va->( $g, @_[1, 0] ) }
+	elsif (@_ == 3) { return $ea->( $g, @_[1, 2, 0] ) }
+	else {
+	    die sprintf "$op: wrong number of arguments (%d)", scalar @_;
+	}
+    } else {
+	die "$op: not a compat02 graph"
+    }
+}
+
+sub _attr02_234 {
+    my ($g, $op, $ga, $va, $ea) = splice @_, 0, 5;
+    if ($g->is_compat02) {
+	if    (@_ == 2) { return $ga->( $g, @_ ) }
+	elsif (@_ == 3) { return $va->( $g, @_[1, 0, 2] ) }
+	elsif (@_ == 4) { return $ea->( $g, @_[1, 2, 0, 3] ) }
+	else {
+	    die sprintf "$op: wrong number of arguments (%d)", scalar @_;
+	}
+    } else {
+	die "$op: not a compat02 graph";
+    }
+}
+
+sub set_attribute {
+    my $g = shift;
+    $g->_attr02_234('set_attribute',
+		    \&Graph::set_graph_attribute,
+		    \&Graph::set_vertex_attribute,
+		    \&Graph::set_edge_attribute,
+		    @_);
+
+}
+
+sub set_attributes {
+    my $g = shift;
+    my $a = pop;
+    $g->_attr02_123('set_attributes',
+		    \&Graph::set_graph_attributes,
+		    \&Graph::set_vertex_attributes,
+		    \&Graph::set_edge_attributes,
+		    $a, @_);
+
+}
+
+sub get_attribute {
+    my $g = shift;
+    $g->_attr02_123('get_attribute',
+		    \&Graph::get_graph_attribute,
+		    \&Graph::get_vertex_attribute,
+		    \&Graph::get_edge_attribute,
+		    @_);
+
+}
+
+sub get_attributes {
+    my $g = shift;
+    $g->_attr02_012('get_attributes',
+		    \&Graph::get_graph_attributes,
+		    \&Graph::get_vertex_attributes,
+		    \&Graph::get_edge_attributes,
+		    @_);
+
+}
+
+sub has_attribute {
+    my $g = shift;
+    return 0 unless @_;
+    $g->_attr02_123('has_attribute',
+		    \&Graph::has_graph_attribute,
+		    \&Graph::has_vertex_attribute,
+		    \&Graph::get_edge_attribute,
+		    @_);
+
+}
+
+sub has_attributes {
+    my $g = shift;
+    $g->_attr02_012('has_attributes',
+		    \&Graph::has_graph_attributes,
+		    \&Graph::has_vertex_attributes,
+		    \&Graph::has_edge_attributes,
+		    @_);
+
+}
+
+sub delete_attribute {
+    my $g = shift;
+    $g->_attr02_123('delete_attribute',
+		    \&Graph::delete_graph_attribute,
+		    \&Graph::delete_vertex_attribute,
+		    \&Graph::delete_edge_attribute,
+		    @_);
+
+}
+
+sub delete_attributes {
+    my $g = shift;
+    $g->_attr02_012('delete_attributes',
+		    \&Graph::delete_graph_attributes,
+		    \&Graph::delete_vertex_attributes,
+		    \&Graph::delete_edge_attributes,
+		    @_);
+
+}
+
+###
+# Simple DFS uses.
+#
+
+sub topological_sort {
+    my $g = shift;
+    my %opt = _get_options( \@_ );
+    my $eic = $opt{ empty_if_cyclic };
+    my $hac;
+    if ($eic) {
+	$hac = $g->has_a_cycle;
+    } else {
+	$g->expect_dag;
+    }
+    delete $opt{ empty_if_cyclic };
+    my $t = Graph::Traversal::DFS->new($g, %opt);
+    my @s = $t->dfs;
+    $hac ? () : reverse @s;
+}
+
+*toposort = \&topological_sort;
+
+sub undirected_copy {
+    my $g = shift;
+
+    $g->expect_directed;
+
+    my $c = Graph::Undirected->new;
+    for my $v ($g->isolated_vertices) { # TODO: if iv ...
+	$c->add_vertex($v);
+    }
+    for my $e ($g->edges05) {
+	$c->add_edge(@$e);
+    }
+    return $c;
+}
+
+*undirected_copy_graph = \&undirected_copy;
+
+sub directed_copy {
+    my $g = shift;
+    $g->expect_undirected;
+    my $c = Graph::Directed->new;
+    for my $v ($g->isolated_vertices) { # TODO: if iv ...
+	$c->add_vertex($v);
+    }
+    for my $e ($g->edges05) {
+	my @e = @$e;
+	$c->add_edge(@e);
+	$c->add_edge(reverse @e);
+    }
+    return $c;
+}
+
+*directed_copy_graph = \&directed_copy;
+
+###
+# Cache or not.
+#
+
+my %_cache_type =
+    (
+     'connectivity'        => '_ccc',
+     'strong_connectivity' => '_scc',
+     'biconnectivity'      => '_bcc',
+     'SPT_Dijkstra'        => '_spt_di',
+     'SPT_Bellman_Ford'    => '_spt_bf',
+    );
+
+sub _check_cache {
+    my ($g, $type, $code) = splice @_, 0, 3;
+    my $c = $_cache_type{$type};
+    if (defined $c) {
+	my $a = $g->get_graph_attribute($c);
+	unless (defined $a && $a->[ 0 ] == $g->[ _G ]) {
+	    $a->[ 0 ] = $g->[ _G ];
+	    $a->[ 1 ] = $code->( $g, @_ );
+	    $g->set_graph_attribute($c, $a);
+	}
+	return $a->[ 1 ];
+    } else {
+	Carp::croak("Graph: unknown cache type '$type'");
+    }
+}
+
+sub _clear_cache {
+    my ($g, $type) = @_;
+    my $c = $_cache_type{$type};
+    if (defined $c) {
+	$g->delete_graph_attribute($c);
+    } else {
+	Carp::croak("Graph: unknown cache type '$type'");
+    }
+}
+
+sub connectivity_clear_cache {
+    my $g = shift;
+    _clear_cache($g, 'connectivity');
+}
+
+sub strong_connectivity_clear_cache {
+    my $g = shift;
+    _clear_cache($g, 'strong_connectivity');
+}
+
+sub biconnectivity_clear_cache {
+    my $g = shift;
+    _clear_cache($g, 'biconnectivity');
+}
+
+sub SPT_Dijkstra_clear_cache {
+    my $g = shift;
+    _clear_cache($g, 'SPT_Dijkstra');
+    $g->delete_graph_attribute('SPT_Dijkstra_first_root');
+}
+
+sub SPT_Bellman_Ford_clear_cache {
+    my $g = shift;
+    _clear_cache($g, 'SPT_Bellman_Ford');
+}
+
+###
+# Connected components.
+#
+
+sub _connected_components_compute {
+    my $g = shift;
+    my %cce;
+    my %cci;
+    my $cc = 0;
+    if ($g->has_union_find) {
+	my $UF = $g->_get_union_find();
+	my $V  = $g->[ _V ];
+	my %icce; # Isolated vertices.
+	my %icci;
+	my $icc = 0;
+	for my $v ( $g->unique_vertices ) {
+	    $cc = $UF->find( $V->_get_path_id( $v ) );
+	    if (defined $cc) {
+		$cce{ $v } = $cc;
+		push @{ $cci{ $cc } }, $v;
+	    } else {
+		$icce{ $v } = $icc;
+		push @{ $icci{ $icc } }, $v;
+		$icc++;
+	    }
+	}
+	if ($icc) {
+	    @cce{ keys %icce } = values %icce;
+	    @cci{ keys %icci } = values %icci;
+	}
+    } else {
+	my @u = $g->unique_vertices;
+	my %r; @r{ @u } = @u;
+	my $froot = sub {
+	    (each %r)[1];
+	};
+	my $nroot = sub {
+	    $cc++ if keys %r;
+	    (each %r)[1];
+	};
+	my $t = Graph::Traversal::DFS->new($g,
+					   first_root => $froot,
+					   next_root  => $nroot,
+					   pre => sub {
+					       my ($v, $t) = @_;
+					       $cce{ $v } = $cc;
+					       push @{ $cci{ $cc } }, $v;
+					       delete $r{ $v };
+					   },
+					   @_);
+	$t->dfs;
+    }
+    return [ \%cce, \%cci ];
+}
+
+sub _connected_components {
+    my $g = shift;
+    my $ccc = _check_cache($g, 'connectivity',
+			   \&_connected_components_compute, @_);
+    return @{ $ccc };
+}
+
+sub connected_component_by_vertex {
+    my ($g, $v) = @_;
+    $g->expect_undirected;
+    my ($CCE, $CCI) = $g->_connected_components();
+    return $CCE->{ $v };
+}
+
+sub connected_component_by_index {
+    my ($g, $i) = @_;
+    $g->expect_undirected;
+    my ($CCE, $CCI) = $g->_connected_components();
+    return defined $CCI->{ $i } ? @{ $CCI->{ $i } } : ( );
+}
+
+sub connected_components {
+    my $g = shift;
+    $g->expect_undirected;
+    my ($CCE, $CCI) = $g->_connected_components();
+    return values %{ $CCI };
+}
+
+sub same_connected_components {
+    my $g = shift;
+    $g->expect_undirected;
+    if ($g->has_union_find) {
+	my $UF = $g->_get_union_find();
+	my $V  = $g->[ _V ];
+	my $u = shift;
+	my $c = $UF->find( $V->_get_path_id ( $u ) );
+	my $d;
+	for my $v ( @_) {
+	    return 0
+		unless defined($d = $UF->find( $V->_get_path_id( $v ) )) &&
+		       $d eq $c;
+	}
+	return 1;
+    } else {
+	my ($CCE, $CCI) = $g->_connected_components();
+	my $u = shift;
+	my $c = $CCE->{ $u };
+	for my $v ( @_) {
+	    return 0
+		unless defined $CCE->{ $v } &&
+		       $CCE->{ $v } eq $c;
+	}
+	return 1;
+    }
+}
+
+my $super_component = sub { join("+", sort @_) };
+
+sub connected_graph {
+    my ($g, %opt) = @_;
+    $g->expect_undirected;
+    my $cg = Graph->new(undirected => 1);
+    if ($g->has_union_find && $g->vertices == 1) {
+	# TODO: super_component?
+	$cg->add_vertices($g->vertices);
+    } else {
+	my $sc_cb =
+	    exists $opt{super_component} ?
+		$opt{super_component} : $super_component;
+	for my $cc ( $g->connected_components() ) {
+	    my $sc = $sc_cb->(@$cc);
+	    $cg->add_vertex($sc);
+	    $cg->set_vertex_attribute($sc, 'subvertices', [ @$cc ]);
+	}
+    }
+    return $cg;
+}
+
+sub is_connected {
+    my $g = shift;
+    $g->expect_undirected;
+    my ($CCE, $CCI) = $g->_connected_components();
+    return keys %{ $CCI } == 1;
+}
+
+sub is_weakly_connected {
+    my $g = shift;
+    $g->expect_directed;
+    $g->undirected_copy->is_connected(@_);
+}
+
+*weakly_connected = \&is_weakly_connected;
+
+sub weakly_connected_components {
+    my $g = shift;
+    $g->expect_directed;
+    $g->undirected_copy->connected_components(@_);
+}
+
+sub weakly_connected_component_by_vertex {
+    my $g = shift;
+    $g->expect_directed;
+    $g->undirected_copy->connected_component_by_vertex(@_);
+}
+
+sub weakly_connected_component_by_index {
+    my $g = shift;
+    $g->expect_directed;
+    $g->undirected_copy->connected_component_by_index(@_);
+}
+
+sub same_weakly_connected_components {
+    my $g = shift;
+    $g->expect_directed;
+    $g->undirected_copy->same_connected_components(@_);
+}
+
+sub weakly_connected_graph {
+    my $g = shift;
+    $g->expect_directed;
+    $g->undirected_copy->connected_graph(@_);
+}
+
+sub _strongly_connected_components_compute {
+    my $g = shift;
+    my $t = Graph::Traversal::DFS->new($g);
+    my @d = reverse $t->dfs;
+    my @c;
+    my $h = $g->transpose_graph;
+    my $u =
+	Graph::Traversal::DFS->new($h,
+				   next_root => sub {
+				       my ($t, $u) = @_;
+				       my $root;
+				       while (defined($root = shift @d)) {
+					   last if exists $u->{ $root };
+				       }
+				       if (defined $root) {
+					   push @c, [];
+					   return $root;
+				       } else {
+					   return;
+				       }
+				   },
+				   pre => sub {
+				       my ($v, $t) = @_;
+				       push @{ $c[-1] }, $v;
+				   },
+				   @_);
+    $u->dfs;
+    return \@c;
+}
+
+sub _strongly_connected_components {
+    my $g = shift;
+    my $scc = _check_cache($g, 'strong_connectivity',
+			   \&_strongly_connected_components_compute, @_);
+    return defined $scc ? @$scc : ( );
+}
+
+sub strongly_connected_components {
+    my $g = shift;
+    $g->expect_directed;
+    $g->_strongly_connected_components(@_);
+}
+
+sub strongly_connected_component_by_vertex {
+    my $g = shift;
+    my $v = shift;
+    $g->expect_directed;
+    my @scc = $g->_strongly_connected_components( next_alphabetic => 1, @_ );
+    for (my $i = 0; $i <= $#scc; $i++) {
+	for (my $j = 0; $j <= $#{ $scc[$i] }; $j++) {
+	    return $i if $scc[$i]->[$j] eq $v;
+	}
+    }
+    return;
+}
+
+sub strongly_connected_component_by_index {
+    my $g = shift;
+    my $i = shift;
+    $g->expect_directed;
+    my $c = ( $g->_strongly_connected_components(@_) )[ $i ];
+    return defined $c ? @{ $c } : ();
+}
+
+sub same_strongly_connected_components {
+    my $g = shift;
+    $g->expect_directed;
+    my @scc = $g->_strongly_connected_components( next_alphabetic => 1, @_ );
+    my @i;
+    while (@_) {
+	my $v = shift;
+	for (my $i = 0; $i <= $#scc; $i++) {
+	    for (my $j = 0; $j <= $#{ $scc[$i] }; $j++) {
+		if ($scc[$i]->[$j] eq $v) {
+		    push @i, $i;
+		    return 0 if @i > 1 && $i[-1] ne $i[0];
+		}
+	    }
+	}
+    }
+    return 1;
+}
+
+sub is_strongly_connected {
+    my $g = shift;
+    $g->expect_directed;
+    my $t = Graph::Traversal::DFS->new($g);
+    my @d = reverse $t->dfs;
+    my @c;
+    my $h = $g->transpose;
+    my $u =
+	Graph::Traversal::DFS->new($h,
+				   next_root => sub {
+				       my ($t, $u) = @_;
+				       my $root;
+				       while (defined($root = shift @d)) {
+					   last if exists $u->{ $root };
+				       }
+				       if (defined $root) {
+					   unless (@{ $t->{ roots } }) {
+					       push @c, [];
+					       return $root;
+					   } else {
+					       $t->terminate;
+					       return;
+					   }
+				       } else {
+					   return;
+				       }
+				   },
+				   pre => sub {
+				       my ($v, $t) = @_;
+				       push @{ $c[-1] }, $v;
+				   },
+				   @_);
+    $u->dfs;
+    return @{ $u->{ roots } } == 1 && keys %{ $u->{ unseen } } == 0;
+}
+
+*strongly_connected = \&is_strongly_connected;
+
+sub strongly_connected_graph {
+    my $g = shift;
+    my %attr = @_;
+
+    $g->expect_directed;
+
+    my $t = Graph::Traversal::DFS->new($g);
+    my @d = reverse $t->dfs;
+    my @c;
+    my $h = $g->transpose;
+    my $u =
+	Graph::Traversal::DFS->new($h,
+				   next_root => sub {
+				       my ($t, $u) = @_;
+				       my $root;
+				       while (defined($root = shift @d)) {
+					   last if exists $u->{ $root };
+				       }
+				       if (defined $root) {
+					   push @c, [];
+					   return $root;
+				       } else {
+					   return;
+				       }
+				   },
+				   pre => sub {
+				       my ($v, $t) = @_;
+				       push @{ $c[-1] }, $v;
+				   }
+				   );
+
+    $u->dfs;
+
+    my $sc_cb;
+    my $hv_cb;
+
+    _opt_get(\%attr, super_component => \$sc_cb);
+    _opt_get(\%attr, hypervertex => \$hv_cb);
+    _opt_unknown(\%attr);
+
+    if (defined $hv_cb && !defined $sc_cb) {
+	$sc_cb = sub { $hv_cb->( [ @_ ] ) };
+    }
+    unless (defined $sc_cb) {
+	$sc_cb = $super_component;
+    }
+
+    my $s = Graph->new;
+
+    my %c;
+    my @s;
+    for (my $i = 0; $i <  @c; $i++) {
+	my $c = $c[$i];
+	$s->add_vertex( $s[$i] = $sc_cb->(@$c) );
+	$s->set_vertex_attribute($s[$i], 'subvertices', [ @$c ]);
+	for my $v (@$c) {
+	    $c{$v} = $i;
+	}
+    }
+
+    my $n = @c;
+    for my $v ($g->vertices) {
+	unless (exists $c{$v}) {
+	    $c{$v} = $n;
+	    $s[$n] = $v;
+	    $n++;
+	}
+    }
+
+    for my $e ($g->edges05) {
+	my ($u, $v) = @$e; # @TODO: hyperedges
+	unless ($c{$u} == $c{$v}) {
+	    my ($p, $q) = ( $s[ $c{ $u } ], $s[ $c{ $v } ] );
+	    $s->add_edge($p, $q) unless $s->has_edge($p, $q);
+	}
+    }
+
+    if (my @i = $g->isolated_vertices) {
+	$s->add_vertices(map { $s[ $c{ $_ } ] } @i);
+    }
+
+    return $s;
+}
+
+###
+# Biconnectivity.
+#
+
+sub _make_bcc {
+    my ($S, $v, $c) = @_;
+    my %b;
+    while (@$S) {
+	my $t = pop @$S;
+	$b{ $t } = $t;
+	last if $t eq $v;
+    }
+    return [ values %b, $c ];
+}
+
+sub _biconnectivity_compute {
+    my $g = shift;
+    my ($opt, $unseenh, $unseena, $r, $next, $code, $attr) =
+	$g->_root_opt(@_);
+    return () unless defined $r;
+    my %P;
+    my %I;
+    for my $v ($g->vertices) {
+	$I{ $v } = 0;
+    }
+    $I{ $r } = 1;
+    my %U;
+    my %S; # Self-loops.
+    for my $e ($g->edges) {
+	my ($u, $v) = @$e;
+	$U{ $u }{ $v } = 0;
+	$U{ $v }{ $u } = 0;
+	$S{ $u } = 1 if $u eq $v;
+    }
+    my $i = 1;
+    my $v = $r;
+    my %AP;
+    my %L = ( $r => 1 );
+    my @S = ( $r );
+    my %A;
+    my @V = $g->vertices;
+
+    # print "V : @V\n";
+    # print "r : $r\n";
+
+    my %T; @T{ @V } = @V;
+
+    for my $w (@V) {
+	my @s = $g->successors( $w );
+	if (@s) {
+	    @s = grep { $_ eq $w ? ( delete $T{ $w }, 0 ) : 1 } @s;
+	    @{ $A{ $w } }{ @s } = @s;
+	} elsif ($g->predecessors( $w ) == 0) {
+	    delete $T{ $w };
+	    if ($w eq $r) {
+		delete $I { $r };
+		$r = $v = each %T;
+		if (defined $r) {
+		    %L = ( $r => 1 );
+		    @S = ( $r );
+		    $I{ $r } = 1;
+		    # print "r : $r\n";
+		}
+	    }
+	}
+    }
+
+    # use Data::Dumper;
+    # print "T : ", Dumper(\%T);
+    # print "A : ", Dumper(\%A);
+
+    my %V2BC;
+    my @BR;
+    my @BC;
+
+    my @C;
+    my $Avok;
+
+    while (keys %T) {
+	# print "T = ", Dumper(\%T);
+	do {
+	    my $w;
+	    do {
+		my @w = _shuffle values %{ $A{ $v } };
+		# print "w = @w\n";
+		$w = first { !$U{ $v }{ $_ } } @w;
+		if (defined $w) {
+		    # print "w = $w\n";
+		    $U{ $v }{ $w }++;
+		    $U{ $w }{ $v }++;
+		    if ($I{ $w } == 0) {
+			$P{ $w } = $v;
+			$i++;
+			$I{ $w } = $i;
+			$L{ $w } = $i;
+			push @S, $w;
+			$v = $w;
+		    } else {
+			$L{ $v } = $I{ $w } if $I{ $w } < $L{ $v };
+		    }
+		}
+	    } while (defined $w);
+	    # print "U = ", Dumper(\%U);
+	    # print "P = ", Dumper(\%P);
+	    # print "L = ", Dumper(\%L);
+	    if (!defined $P{ $v }) {
+		# Do nothing.
+	    } elsif ($P{ $v } ne $r) {
+		if ($L{ $v } < $I{ $P{ $v } }) {
+		    $L{ $P{ $v } } = $L{ $v } if $L{ $v } < $L{ $P{ $v } };
+		} else {
+		    $AP{ $P{ $v } } = $P{ $v };
+		    push @C, _make_bcc(\@S, $v, $P{ $v } );
+		}
+	    } else {
+		my $e;
+		for my $w (_shuffle keys %{ $A{ $r } }) {
+		    # print "w = $w\n";
+		    unless ($U{ $r }{ $w }) {
+			$e = $r;
+			# print "e = $e\n";
+			last;
+		    }
+		}
+		$AP{ $e } = $e if defined $e;
+		push @C, _make_bcc(\@S, $v, $r);
+	    }
+	    # print "AP = ", Dumper(\%AP);
+	    # print "C  = ", Dumper(\@C);
+	    # print "L  = ", Dumper(\%L);
+	    $v = defined $P{ $v } ? $P{ $v } : $r;
+	    # print "v = $v\n";
+	    $Avok = 0;
+	    if (defined $v) {
+		if (keys %{ $A{ $v } }) {
+		    if (!exists $P{ $v }) {
+			for my $w (keys %{ $A{ $v } }) {
+			    $Avok++ if $U{ $v }{ $w };
+			}
+			# print "Avok/1 = $Avok\n";
+			$Avok = 0 unless $Avok == keys %{ $A{ $v } };
+			# print "Avok/2 = $Avok\n";
+		    }
+		} else {
+		    $Avok = 1;
+		    # print "Avok/3 = $Avok\n";
+		}
+	    }
+	} until ($Avok);
+
+	last if @C == 0 && !exists $S{$v};
+
+	for (my $i = 0; $i < @C; $i++) {
+	    for my $v (@{ $C[ $i ]}) {
+		$V2BC{ $v }{ $i }++;
+		delete $T{ $v };
+	    }
+	}
+
+	for (my $i = 0; $i < @C; $i++) {
+	    if (@{ $C[ $i ] } == 2) {
+		push @BR, $C[ $i ];
+	    } else {
+		push @BC, $C[ $i ];
+	    }
+	}
+
+	if (keys %T) {
+	    $r = $v = each %T;
+	}
+    }
+    
+    return [ [values %AP], \@BC, \@BR, \%V2BC ];
+}
+
+sub biconnectivity {
+    my $g = shift;
+    $g->expect_undirected;
+    my $bcc = _check_cache($g, 'biconnectivity',
+			   \&_biconnectivity_compute, @_);
+    return defined $bcc ? @$bcc : ( );
+}
+
+sub is_biconnected {
+    my $g = shift;
+    my ($ap, $bc) = ($g->biconnectivity(@_))[0, 1];
+    return defined $ap ? @$ap == 0 && $g->vertices >= 3 : undef;
+}
+
+sub is_edge_connected {
+    my $g = shift;
+    my ($br) = ($g->biconnectivity(@_))[2];
+    return defined $br ? @$br == 0 && $g->edges : undef;
+}
+
+sub is_edge_separable {
+    my $g = shift;
+    my $c = $g->is_edge_connected;
+    defined $c ? !$c && $g->edges : undef;
+}
+
+sub articulation_points {
+    my $g = shift;
+    my ($ap) = ($g->biconnectivity(@_))[0];
+    return defined $ap ? @$ap : ();
+}
+
+*cut_vertices = \&articulation_points;
+
+sub biconnected_components {
+    my $g = shift;
+    my ($bc) = ($g->biconnectivity(@_))[1];
+    return defined $bc ? @$bc : ();
+}
+
+sub biconnected_component_by_index {
+    my $g = shift;
+    my $i = shift;
+    my ($bc) = ($g->biconnectivity(@_))[1];
+    return defined $bc ? $bc->[ $i ] : undef;
+}
+
+sub biconnected_component_by_vertex {
+    my $g = shift;
+    my $v = shift;
+    my ($v2bc) = ($g->biconnectivity(@_))[3];
+    return defined $v2bc->{ $v } ? keys %{ $v2bc->{ $v } } : ();
+}
+
+sub same_biconnected_components {
+    my $g = shift;
+    my $u = shift;
+    my @u = $g->biconnected_component_by_vertex($u, @_);
+    return 0 unless @u;
+    my %ubc; @ubc{ @u } = ();
+    while (@_) {
+	my $v = shift;
+	my @v = $g->biconnected_component_by_vertex($v);
+	if (@v) {
+	    my %vbc; @vbc{ @v } = ();
+	    my $vi;
+	    for my $ui (keys %ubc) {
+		if (exists $vbc{ $ui }) {
+		    $vi = $ui;
+		    last;
+		}
+	    }
+	    return 0 unless defined $vi;
+	}
+    }
+    return 1;
+}
+
+sub biconnected_graph {
+    my ($g, %opt) = @_;
+    my ($bc, $v2bc) = ($g->biconnectivity, %opt)[1, 3];
+    my $bcg = Graph::Undirected->new;
+    my $sc_cb =
+	exists $opt{super_component} ?
+	    $opt{super_component} : $super_component;
+    for my $c (@$bc) {
+	$bcg->add_vertex(my $s = $sc_cb->(@$c));
+	$bcg->set_vertex_attribute($s, 'subvertices', [ @$c ]);
+    }
+    my %k;
+    for my $i (0..$#$bc) {
+	my @u = @{ $bc->[ $i ] };
+	my %i; @i{ @u } = ();
+	for my $j (0..$#$bc) {
+	    if ($i > $j) {
+		my @v = @{ $bc->[ $j ] };
+		my %j; @j{ @v } = ();
+		for my $u (@u) {
+		    if (exists $j{ $u }) {
+			unless ($k{ $i }{ $j }++) {
+			    $bcg->add_edge($sc_cb->(@{$bc->[$i]}),
+					   $sc_cb->(@{$bc->[$j]}));
+			}
+			last;
+		    }
+		}
+	    }
+	}
+    }
+    return $bcg;
+}
+
+sub bridges {
+    my $g = shift;
+    my ($br) = ($g->biconnectivity(@_))[2];
+    return defined $br ? @$br : ();
+}
+
+###
+# SPT.
+#
+
+sub _SPT_add {
+    my ($g, $h, $HF, $r, $attr, $unseen, $etc) = @_;
+    my $etc_r = $etc->{ $r } || 0;
+    for my $s ( grep { exists $unseen->{ $_ } } $g->successors( $r ) ) {
+	my $t = $g->get_edge_attribute( $r, $s, $attr );
+	$t = 1 unless defined $t;
+	if ($t < 0) {
+	    require Carp;
+	    Carp::croak("Graph::SPT_Dijkstra: edge $r-$s is negative ($t)");
+	}
+	if (!defined($etc->{ $s }) || ($etc_r + $t) < $etc->{ $s }) {
+	    my $etc_s = $etc->{ $s } || 0;
+	    $etc->{ $s } = $etc_r + $t;
+	    # print "$r - $s : setting $s to $etc->{ $s } ($etc_r, $etc_s)\n";
+	    $h->set_vertex_attribute( $s, $attr, $etc->{ $s });
+	    $h->set_vertex_attribute( $s, 'p', $r );
+	    $HF->add( Graph::SPTHeapElem->new($r, $s, $etc->{ $s }) );
+	}
+    }
+}
+
+sub _SPT_Dijkstra_compute {
+}
+
+sub SPT_Dijkstra {
+    my $g = shift;
+    my %opt = @_ == 1 ? (first_root => $_[0]) : @_;
+    my $first_root = $opt{ first_root };
+    unless (defined $first_root) {
+	$opt{ first_root } = $first_root = $g->random_vertex();
+    }
+    my $spt_di = $g->get_graph_attribute('_spt_di');
+    unless (defined $spt_di && exists $spt_di->{ $first_root } && $spt_di->{ $first_root }->[ 0 ] == $g->[ _G ]) {
+	my %etc;
+	my $sptg = $g->_heap_walk($g->new, \&_SPT_add, \%etc, %opt);
+	$spt_di->{ $first_root } = [ $g->[ _G ], $sptg ];
+	$g->set_graph_attribute('_spt_di', $spt_di);
+    }
+
+    my $spt = $spt_di->{ $first_root }->[ 1 ];
+
+    $spt->set_graph_attribute('SPT_Dijkstra_root', $first_root);
+
+    return $spt;
+}
+
+*SSSP_Dijkstra = \&SPT_Dijkstra;
+
+*single_source_shortest_paths = \&SPT_Dijkstra;
+
+sub SP_Dijkstra {
+    my ($g, $u, $v) = @_;
+    my $sptg = $g->SPT_Dijkstra(first_root => $u);
+    my @path = ($v);
+    my %seen;
+    my $V = $g->vertices;
+    my $p;
+    while (defined($p = $sptg->get_vertex_attribute($v, 'p'))) {
+	last if exists $seen{$p};
+	push @path, $p;
+	$v = $p;
+	$seen{$p}++;
+	last if keys %seen == $V || $u eq $v;
+    }
+    @path = () if @path && $path[-1] ne $u;
+    return reverse @path;
+}
+
+sub __SPT_Bellman_Ford {
+    my ($g, $u, $v, $attr, $d, $p, $c0, $c1) = @_;
+    return unless $c0->{ $u };
+    my $w = $g->get_edge_attribute($u, $v, $attr);
+    $w = 1 unless defined $w;
+    if (defined $d->{ $v }) {
+	if (defined $d->{ $u }) {
+	    if ($d->{ $v } > $d->{ $u } + $w) {
+		$d->{ $v } = $d->{ $u } + $w;
+		$p->{ $v } = $u;
+		$c1->{ $v }++;
+	    }
+	} # else !defined $d->{ $u } &&  defined $d->{ $v }
+    } else {
+	if (defined $d->{ $u }) {
+	    #  defined $d->{ $u } && !defined $d->{ $v }
+	    $d->{ $v } = $d->{ $u } + $w;
+	    $p->{ $v } = $u;
+	    $c1->{ $v }++;
+	} # else !defined $d->{ $u } && !defined $d->{ $v }
+    }
+}
+
+sub _SPT_Bellman_Ford {
+    my ($g, $opt, $unseenh, $unseena, $r, $next, $code, $attr) = @_;
+    my %d;
+    return unless defined $r;
+    $d{ $r } = 0;
+    my %p;
+    my $V = $g->vertices;
+    my %c0; # Changed during the last iteration?
+    $c0{ $r }++;
+    for (my $i = 0; $i < $V; $i++) {
+	my %c1;
+	for my $e ($g->edges) {
+	    my ($u, $v) = @$e;
+	    __SPT_Bellman_Ford($g, $u, $v, $attr, \%d, \%p, \%c0, \%c1);
+	    if ($g->undirected) {
+		__SPT_Bellman_Ford($g, $v, $u, $attr, \%d, \%p, \%c0, \%c1);
+	    }
+	}
+	%c0 = %c1 unless $i == $V - 1;
+    }
+
+    for my $e ($g->edges) {
+	my ($u, $v) = @$e;
+	if (defined $d{ $u } && defined $d{ $v }) {
+	    my $d = $g->get_edge_attribute($u, $v, $attr);
+	    if (defined $d && $d{ $v } > $d{ $u } + $d) {
+		require Carp;
+		Carp::croak("Graph::SPT_Bellman_Ford: negative cycle exists");
+	    }
+	}
+    }
+
+    return (\%p, \%d);
+}
+
+sub _SPT_Bellman_Ford_compute {
+}
+
+sub SPT_Bellman_Ford {
+    my $g = shift;
+
+    my ($opt, $unseenh, $unseena, $r, $next, $code, $attr) = $g->_root_opt(@_);
+
+    unless (defined $r) {
+	$r = $g->random_vertex();
+	return unless defined $r;
+    }
+
+    my $spt_bf = $g->get_graph_attribute('_spt_bf');
+    unless (defined $spt_bf &&
+	    exists $spt_bf->{ $r } && $spt_bf->{ $r }->[ 0 ] == $g->[ _G ]) {
+	my ($p, $d) =
+	    $g->_SPT_Bellman_Ford($opt, $unseenh, $unseena,
+				  $r, $next, $code, $attr);
+	my $h = $g->new;
+	for my $v (keys %$p) {
+	    my $u = $p->{ $v };
+	    $h->add_edge( $u, $v );
+	    $h->set_edge_attribute( $u, $v, $attr,
+				    $g->get_edge_attribute($u, $v, $attr));
+	    $h->set_vertex_attribute( $v, $attr, $d->{ $v } );
+	    $h->set_vertex_attribute( $v, 'p', $u );
+	}
+	$spt_bf->{ $r } = [ $g->[ _G ], $h ];
+	$g->set_graph_attribute('_spt_bf', $spt_bf);
+    }
+
+    my $spt = $spt_bf->{ $r }->[ 1 ];
+
+    $spt->set_graph_attribute('SPT_Bellman_Ford_root', $r);
+
+    return $spt;
+}
+
+*SSSP_Bellman_Ford = \&SPT_Bellman_Ford;
+
+sub SP_Bellman_Ford {
+    my ($g, $u, $v) = @_;
+    my $sptg = $g->SPT_Bellman_Ford(first_root => $u);
+    my @path = ($v);
+    my %seen;
+    my $V = $g->vertices;
+    my $p;
+    while (defined($p = $sptg->get_vertex_attribute($v, 'p'))) {
+	last if exists $seen{$p};
+	push @path, $p;
+	$v = $p;
+	$seen{$p}++;
+	last if keys %seen == $V;
+    }
+    # @path = () if @path && "$path[-1]" ne "$u";
+    return reverse @path;
+}
+
+###
+# Transitive Closure.
+#
+
+sub TransitiveClosure_Floyd_Warshall {
+    my $self = shift;
+    my $class = ref $self || $self;
+    $self = shift unless ref $self;
+    bless Graph::TransitiveClosure->new($self, @_), $class;
+}
+
+*transitive_closure = \&TransitiveClosure_Floyd_Warshall;
+
+sub APSP_Floyd_Warshall {
+    my $self = shift;
+    my $class = ref $self || $self;
+    $self = shift unless ref $self;
+    bless Graph::TransitiveClosure->new($self, path => 1, @_), $class;
+}
+
+*all_pairs_shortest_paths = \&APSP_Floyd_Warshall;
+
+sub _transitive_closure_matrix_compute {
+}
+
+sub transitive_closure_matrix {
+    my $g = shift;
+    my $tcm = $g->get_graph_attribute('_tcm');
+    if (defined $tcm) {
+	if (ref $tcm eq 'ARRAY') { # YECHHH!
+	    if ($tcm->[ 0 ] == $g->[ _G ]) {
+		$tcm = $tcm->[ 1 ];
+	    } else {
+		undef $tcm;
+	    }
+	}
+    }
+    unless (defined $tcm) {
+	my $apsp = $g->APSP_Floyd_Warshall(@_);
+	$tcm = $apsp->get_graph_attribute('_tcm');
+	$g->set_graph_attribute('_tcm', [ $g->[ _G ], $tcm ]);
+    }
+
+    return $tcm;
+}
+
+sub path_length {
+    my $g = shift;
+    my $tcm = $g->transitive_closure_matrix;
+    $tcm->path_length(@_);
+}
+
+sub path_predecessor {
+    my $g = shift;
+    my $tcm = $g->transitive_closure_matrix;
+    $tcm->path_predecessor(@_);
+}
+
+sub path_vertices {
+    my $g = shift;
+    my $tcm = $g->transitive_closure_matrix;
+    $tcm->path_vertices(@_);
+}
+
+sub is_reachable {
+    my $g = shift;
+    my $tcm = $g->transitive_closure_matrix;
+    $tcm->is_reachable(@_);
+}
+
+sub for_shortest_paths {
+    my $g = shift;
+    my $c = shift;
+    my $t = $g->transitive_closure_matrix;
+    my @v = $g->vertices;
+    my $n = 0;
+    for my $u (@v) {
+	for my $v (@v) {
+	    next unless $t->is_reachable($u, $v);
+	    $n++;
+	    $c->($t, $u, $v, $n);
+	}
+    }
+    return $n;
+}
+
+sub _minmax_path {
+    my $g = shift;
+    my $min;
+    my $max;
+    my $minp;
+    my $maxp;
+    $g->for_shortest_paths(sub {
+			       my ($t, $u, $v, $n) = @_;
+			       my $l = $t->path_length($u, $v);
+			       return unless defined $l;
+			       my $p;
+			       if ($u ne $v && (!defined $max || $l > $max)) {
+				   $max = $l;
+				   $maxp = $p = [ $t->path_vertices($u, $v) ];
+			       }
+			       if ($u ne $v && (!defined $min || $l < $min)) {
+				   $min = $l;
+				   $minp = $p || [ $t->path_vertices($u, $v) ];
+			       }
+			   });
+    return ($min, $max, $minp, $maxp);
+}
+
+sub diameter {
+    my $g = shift;
+    my ($min, $max, $minp, $maxp) = $g->_minmax_path(@_);
+    return defined $maxp ? (wantarray ? @$maxp : $max) : undef;
+}
+
+*graph_diameter = \&diameter;
+
+sub longest_path {
+    my ($g, $u, $v) = @_;
+    my $t = $g->transitive_closure_matrix;
+    if (defined $u) {
+	if (defined $v) {
+	    return wantarray ?
+		$t->path_vertices($u, $v) : $t->path_length($u, $v);
+	} else {
+	    my $max;
+	    my @max;
+	    for my $v ($g->vertices) {
+		next if $u eq $v;
+		my $l = $t->path_length($u, $v);
+		if (defined $l && (!defined $max || $l > $max)) {
+		    $max = $l;
+		    @max = $t->path_vertices($u, $v);
+		}
+	    }
+	    return wantarray ? @max : $max;
+	}
+    } else {
+	if (defined $v) {
+	    my $max;
+	    my @max;
+	    for my $u ($g->vertices) {
+		next if $u eq $v;
+		my $l = $t->path_length($u, $v);
+		if (defined $l && (!defined $max || $l > $max)) {
+		    $max = $l;
+		    @max = $t->path_vertices($u, $v);
+		}
+	    }
+	    return wantarray ? @max : @max - 1;
+	} else {
+	    my ($min, $max, $minp, $maxp) = $g->_minmax_path(@_);
+	    return defined $maxp ? (wantarray ? @$maxp : $max) : undef;
+	}
+    }
+}
+
+sub vertex_eccentricity {
+    my ($g, $u) = @_;
+    $g->expect_undirected;
+    if ($g->is_connected) {
+	my $max;
+	for my $v ($g->vertices) {
+	    next if $u eq $v;
+	    my $l = $g->path_length($u, $v);
+	    if (defined $l && (!defined $max || $l > $max)) {
+		$max = $l;
+	    }
+	}
+	return $max;
+    } else {
+	return Infinity();
+    }
+}
+
+sub shortest_path {
+    my ($g, $u, $v) = @_;
+    $g->expect_undirected;
+    my $t = $g->transitive_closure_matrix;
+    if (defined $u) {
+	if (defined $v) {
+	    return wantarray ?
+		$t->path_vertices($u, $v) : $t->path_length($u, $v);
+	} else {
+	    my $min;
+	    my @min;
+	    for my $v ($g->vertices) {
+		next if $u eq $v;
+		my $l = $t->path_length($u, $v);
+		if (defined $l && (!defined $min || $l < $min)) {
+		    $min = $l;
+		    @min = $t->path_vertices($u, $v);
+		}
+	    }
+	    return wantarray ? @min : $min;
+	}
+    } else {
+	if (defined $v) {
+	    my $min;
+	    my @min;
+	    for my $u ($g->vertices) {
+		next if $u eq $v;
+		my $l = $t->path_length($u, $v);
+		if (defined $l && (!defined $min || $l < $min)) {
+		    $min = $l;
+		    @min = $t->path_vertices($u, $v);
+		}
+	    }
+	    return wantarray ? @min : $min;
+	} else {
+	    my ($min, $max, $minp, $maxp) = $g->_minmax_path(@_);
+	    return defined $minp ? (wantarray ? @$minp : $min) : undef;
+	}
+    }
+}
+
+sub radius {
+    my $g = shift;
+    $g->expect_undirected;
+    my ($center, $radius) = (undef, Infinity());
+    for my $v ($g->vertices) {
+	my $x = $g->vertex_eccentricity($v);
+	($center, $radius) = ($v, $x) if defined $x && $x < $radius;
+    }
+    return $radius;
+}
+
+sub center_vertices {
+    my ($g, $delta) = @_;
+    $g->expect_undirected;
+    $delta = 0 unless defined $delta;
+    $delta = abs($delta);
+    my @c;
+    my $r = $g->radius;
+    if (defined $r) {
+	for my $v ($g->vertices) {
+	    my $e = $g->vertex_eccentricity($v);
+	    next unless defined $e;
+	    push @c, $v if abs($e - $r) <= $delta;
+	}
+    }
+    return @c;
+}
+
+*centre_vertices = \&center_vertices;
+
+sub average_path_length {
+    my $g = shift;
+    my @A = @_;
+    my $d = 0;
+    my $m = 0;
+    my $n = $g->for_shortest_paths(sub {
+				       my ($t, $u, $v, $n) = @_;
+				       my $l = $t->path_length($u, $v);
+				       if ($l) {
+					   my $c = @A == 0 ||
+					       (@A == 1 && $u eq $A[0]) ||
+						   ((@A == 2) &&
+						    (defined $A[0] &&
+						     $u eq $A[0]) ||
+						    (defined $A[1] &&
+						     $v eq $A[1]));
+					   if ($c) {
+					       $d += $l;
+					       $m++;
+					   }
+				       }
+				   });
+    return $m ? $d / $m : undef;
+}
+
+###
+# Simple tests.
+#
+
+sub is_multi_graph {
+    my $g = shift;
+    return 0 unless $g->is_multiedged || $g->is_countedged;
+    my $multiedges = 0;
+    for my $e ($g->edges05) {
+	my ($u, @v) = @$e;
+	for my $v (@v) {
+	    return 0 if $u eq $v;
+	}
+	$multiedges++ if $g->get_edge_count(@$e) > 1;
+    }
+    return $multiedges;
+}
+
+sub is_simple_graph {
+    my $g = shift;
+    return 1 unless $g->is_countedged || $g->is_multiedged;
+    for my $e ($g->edges05) {
+	return 0 if $g->get_edge_count(@$e) > 1;
+    }
+    return 1;
+}
+
+sub is_pseudo_graph {
+    my $g = shift;
+    my $m = $g->is_countedged || $g->is_multiedged;
+    for my $e ($g->edges05) {
+	my ($u, @v) = @$e;
+	for my $v (@v) {
+	    return 1 if $u eq $v;
+	}
+	return 1 if $m && $g->get_edge_count($u, @v) > 1;
+    }
+    return 0;
+}
+
+###
+# Rough isomorphism guess.
+#
+
+my %_factorial = (0 => 1, 1 => 1);
+
+sub __factorial {
+    my $n = shift;
+    for (my $i = 2; $i <= $n; $i++) {
+	next if exists $_factorial{$i};
+	$_factorial{$i} = $i * $_factorial{$i - 1};
+    }
+    $_factorial{$n};
+}
+
+sub _factorial {
+    my $n = int(shift);
+    if ($n < 0) {
+	require Carp;
+	Carp::croak("factorial of a negative number");
+    }
+    __factorial($n) unless exists $_factorial{$n};
+    return $_factorial{$n};
+}
+
+sub could_be_isomorphic {
+    my ($g0, $g1) = @_;
+    return 0 unless $g0->vertices == $g1->vertices;
+    return 0 unless $g0->edges05  == $g1->edges05;
+    my %d0;
+    for my $v0 ($g0->vertices) {
+	$d0{ $g0->in_degree($v0) }{ $g0->out_degree($v0) }++
+    }
+    my %d1;
+    for my $v1 ($g1->vertices) {
+	$d1{ $g1->in_degree($v1) }{ $g1->out_degree($v1) }++
+    }
+    return 0 unless keys %d0 == keys %d1;
+    for my $da (keys %d0) {
+	return 0
+	    unless exists $d1{$da} &&
+		   keys %{ $d0{$da} } == keys %{ $d1{$da} };
+	for my $db (keys %{ $d0{$da} }) {
+	    return 0
+		unless exists $d1{$da}{$db} && 
+		       $d0{$da}{$db} == $d1{$da}{$db};
+	}
+    }
+    for my $da (keys %d0) {
+	for my $db (keys %{ $d0{$da} }) {
+	    return 0 unless $d1{$da}{$db} == $d0{$da}{$db};
+	}
+	delete $d1{$da};
+    }
+    return 0 unless keys %d1 == 0;
+    my $f = 1;
+    for my $da (keys %d0) {
+	for my $db (keys %{ $d0{$da} }) {
+	    $f *= _factorial(abs($d0{$da}{$db}));
+	}
+    }
+    return $f;
+}
+
+###
+# Debugging.
+#
+
+sub _dump {
+    require Data::Dumper;
+    my $d = Data::Dumper->new([$_[0]],[ref $_[0]]);
+    defined wantarray ? $d->Dump : print $d->Dump;
+}
+
+1;
diff --git a/perllib/Graph.pod b/perllib/Graph.pod
new file mode 100644
index 00000000..9452d51d
--- /dev/null
+++ b/perllib/Graph.pod
@@ -0,0 +1,2768 @@
+=pod
+
+=head1 NAME
+
+Graph - graph data structures and algorithms
+
+=head1 SYNOPSIS
+
+	use Graph;
+	my $g0 = Graph->new;             # A directed graph.
+
+	use Graph::Directed;
+	my $g1 = Graph::Directed->new;   # A directed graph.
+
+	use Graph::Undirected;
+	my $g2 = Graph::Undirected->new; # An undirected graph.
+
+        $g->add_edge(...);
+        $g->has_edge(...)
+        $g->delete_edge(...);
+
+        $g->add_vertex(...);
+        $g->has_vertex(...);
+        $g->delete_vertex(...);
+
+        $g->vertices(...)
+        $g->edges(...)
+
+        # And many, many more, see below.
+
+=head1 DESCRIPTION
+
+=head2 Non-Description
+
+This module is not for B<drawing> any sort of I<graphics>, business or
+otherwise.
+
+=head2 Description
+
+Instead, this module is for creating I<abstract data structures>
+called graphs, and for doing various operations on those.
+
+=head2 Perl 5.6.0 minimum
+
+The implementation depends on a Perl feature called "weak references"
+and Perl 5.6.0 was the first to have those.
+
+=head2 Constructors
+
+=over 4
+
+=item new
+
+Create an empty graph.
+
+=item Graph->new(%options)
+
+The options are a hash with option names as the hash keys and the option
+values as the hash values.
+
+The following options are available:
+
+=over 8
+
+=item *
+
+directed
+
+A boolean option telling that a directed graph should be created.
+Often somewhat redundant because a directed graph is the default
+for the Graph class or one could simply use the C<new()> constructor
+of the Graph::Directed class.
+
+You can test the directness of a graph with $g->is_directed() and
+$g->is_undirected().
+
+=item *
+
+undirected
+
+A boolean option telling that an undirected graph should be created.
+One could also use the C<new()> constructor the Graph::Undirected class
+instead.
+
+Note that while often it is possible to think undirected graphs as
+bidirectional graphs, or as directed graphs with edges going both ways,
+in this module directed graphs and undirected graphs are two different
+things that often behave differently.
+
+You can test the directness of a graph with $g->is_directed() and
+$g->is_undirected().
+
+=item *
+
+refvertexed
+
+If you want to use references (including Perl objects) as vertices.
+
+=item *
+
+unionfind
+
+If the graph is undirected, you can specify the C<unionfind> parameter
+to use the so-called union-find scheme to speed up the computation of
+I<connected components> of the graph (see L</is_connected>,
+L</connected_components>, L</connected_component_by_vertex>,
+L</connected_component_by_index>, and L</same_connected_components>).
+If C<unionfind> is used, adding edges (and vertices) becomes slower,
+but connectedness queries become faster.  You can test a graph for
+"union-findness" with
+
+=over 8
+
+=item has_union_find
+
+    has_union_find
+
+=back
+
+=item *
+
+vertices
+
+An array reference of vertices to add.
+
+=item *
+
+edges
+
+An array reference of array references of edge vertices to add.
+
+=back
+
+=item copy
+
+=item copy_graph
+
+    my $c = $g->copy_graph;
+
+Create a shallow copy of the structure (vertices and edges) of the graph.
+If you want a deep copy that includes attributes, see L</deep_copy>.
+The copy will have the same directedness as the original.
+
+=item deep_copy
+
+=item deep_copy_graph
+
+    my $c = $g->deep_copy_graph;
+
+Create a deep copy of the graph (vertices, edges, and attributes) of
+the graph.  If you want a shallow copy that does not include attributes,
+see L</copy>.  (Uses Data::Dumper behind the scenes.  Note that copying
+code references only works with Perls 5.8 or later, and even then only
+if B::Deparse can reconstruct your code.)
+
+=item undirected_copy
+
+=item undirected_copy_graph
+
+    my $c = $g->undirected_copy_graph;
+
+Create an undirected shallow copy (vertices and edges) of the directed graph
+so that for any directed edge (u, v) there is an undirected edge (u, v).
+
+=item directed_copy
+
+=item directed_copy_graph
+
+    my $c = $g->directed_copy_graph;
+
+Create a directed shallow copy (vertices and edges) of the undirected graph
+so that for any undirected edge (u, v) there are two directed edges (u, v)
+and (v, u).
+
+=item transpose
+
+=item transpose_graph
+
+    my $t = $g->transpose_graph;
+
+Create a directed shallow transposed copy (vertices and edges) of the
+directed graph so that for any directed edge (u, v) there is a directed
+edge (v, u).
+
+You can also transpose a single edge with
+
+=over 8
+
+=item transpose_edge
+
+    $g->transpose_edge($u, $v)
+
+=back
+
+=item complete_graph
+
+=item complete
+
+    my $c = $g->complete_graph;
+
+Create a complete graph that has the same vertices as the original graph.
+A complete graph has an edge between every pair of vertices.
+
+=item complement_graph
+
+=item complement
+
+    my $c = $g->complement_graph;
+
+Create a complement graph that has the same vertices as the original graph.
+A complement graph has an edge (u,v) if and only if the original
+graph does not have edge (u,v).
+
+=back
+
+See also L</random_graph> for a random constructor.
+
+=head2 Basics
+
+=over 4
+
+=item add_vertex
+
+    $g->add_vertex($v)
+
+Add the vertex to the graph.  Returns the graph.
+
+By default idempotent, but a graph can be created I<countvertexed>.
+
+A vertex is also known as a I<node>.
+
+Adding C<undef> as vertex is not allowed.
+
+Note that unless you have isolated vertices (or I<countvertexed>
+vertices), you do not need to explicitly use C<add_vertex> since
+L</add_edge> will implicitly add its vertices.
+
+=item add_edge
+
+    $g->add_edge($u, $v)
+
+Add the edge to the graph.  Implicitly first adds the vertices if the
+graph does not have them.  Returns the graph.
+
+By default idempotent, but a graph can be created I<countedged>.
+
+An edge is also known as an I<arc>.
+
+=item has_vertex
+
+    $g->has_vertex($v)
+
+Return true if the vertex exists in the graph, false otherwise.
+
+=item has_edge
+
+    $g->has_edge($u, $v)
+
+Return true if the edge exists in the graph, false otherwise.
+
+=item delete_vertex
+
+    $g->delete_vertex($v)
+
+Delete the vertex from the graph.  Returns the graph, even
+if the vertex did not exist in the graph.
+
+If the graph has been created I<multivertexed> or I<countvertexed>
+and a vertex has been added multiple times, the vertex will require
+at least an equal number of deletions to become completely deleted.
+
+=item delete_vertices
+
+    $g->delete_vertices($v1, $v2, ...)
+
+Delete the vertices from the graph.  Returns the graph.
+
+If the graph has been created I<multivertexed> or I<countvertexed>
+and a vertex has been added multiple times, the vertex will require
+at least an equal number of deletions to become completely deleteted.
+
+=item delete_edge
+
+    $g->delete_edge($u, $v)
+
+Delete the edge from the graph.  Returns the graph, even
+if the edge did not exist in the graph.
+
+If the graph has been created I<multivertexed> or I<countedged>
+and an edge has been added multiple times, the edge will require
+at least an equal number of deletions to become completely deleted.
+
+=item delete_edges
+
+    $g->delete_edges($u1, $v1, $u2, $v2, ...)
+
+Delete the edges from the graph.  Returns the graph.
+
+If the graph has been created I<multivertexed> or I<countedged>
+and an edge has been added multiple times, the edge will require
+at least an equal number of deletions to become completely deleted.
+
+=back
+
+=head2 Displaying
+
+Graphs have stringification overload, so you can do things like
+
+    print "The graph is $g\n"
+
+One-way (directed, unidirected) edges are shown as '-', two-way
+(undirected, bidirected) edges are shown as '='.  If you want to,
+you can call the stringification via the method
+
+=over 4
+
+=item stringify
+
+=back
+
+=head2 Comparing
+
+Testing for equality can be done either by the overloaded C<eq>
+operator
+
+    $g eq "a-b,a-c,d"
+
+or by the method
+
+=over 4
+
+=item eq
+
+    $g->eq("a-b,a-c,d")
+
+=back
+
+The equality testing compares the stringified forms, and therefore it
+assumes total equality, not isomorphism: all the vertices must be
+named the same, and they must have identical edges between them.
+
+For unequality there are correspondingly the overloaded C<ne>
+operator and the method
+
+=over 4
+
+=item ne
+
+    $g->ne("a-b,a-c,d")
+
+=back
+
+See also L</Isomorphism>.
+
+=head2 Paths and Cycles
+
+Paths and cycles are simple extensions of edges: paths are edges
+starting from where the previous edge ended, and cycles are paths
+returning back to the start vertex of the first edge.
+
+=over 4
+
+=item add_path
+
+   $g->add_path($a, $b, $c, ..., $x, $y, $z)
+
+Add the edges $a-$b, $b-$c, ..., $x-$y, $y-$z to the graph.
+Returns the graph.
+
+=item has_path
+
+   $g->has_path($a, $b, $c, ..., $x, $y, $z)
+
+Return true if the graph has all the edges $a-$b, $b-$c, ..., $x-$y, $y-$z,
+false otherwise.
+
+=item delete_path
+
+   $g->delete_path($a, $b, $c, ..., $x, $y, $z)
+
+Delete all the edges edges $a-$b, $b-$c, ..., $x-$y, $y-$z
+(regardless of whether they exist or not).  Returns the graph.
+
+=item add_cycle
+
+   $g->add_cycle($a, $b, $c, ..., $x, $y, $z)
+
+Add the edges $a-$b, $b-$c, ..., $x-$y, $y-$z, and $z-$a to the graph.
+Returns the graph.
+
+=item has_cycle
+
+   $g->has_cycle($a, $b, $c, ..., $x, $y, $z)
+
+Return true if the graph has all the edges $a-$b, $b-$c, ..., $x-$y, $y-$z,
+and $z-$a, false otherwise.
+
+B<NOTE:> This does not I<detect> cycles, see L</has_a_cycle> and
+L</find_a_cycle>.
+
+=item delete_cycle
+
+   $g->delete_cycle($a, $b, $c, ..., $x, $y, $z)
+
+Delete all the edges edges $a-$b, $b-$c, ..., $x-$y, $y-$z, and $z-$a
+(regardless of whether they exist or not).  Returns the graph.
+
+=item has_a_cycle
+
+   $g->has_a_cycle
+
+Returns true if the graph has a cycle, false if not.
+
+=item find_a_cycle
+
+   $g->find_a_cycle
+
+Returns a cycle if the graph has one (as a list of vertices), an empty
+list if no cycle can be found.
+
+Note that this just returns the vertices of I<a cycle>: not any
+particular cycle, just the first one it finds.  A repeated call
+might find the same cycle, or it might find a different one, and
+you cannot call this repeatedly to find all the cycles.
+
+=back
+
+=head2 Graph Types
+
+=over 4
+
+=item is_simple_graph
+
+    $g->is_simple_graph
+
+Return true if the graph has no multiedges, false otherwise.
+
+=item is_pseudo_graph
+
+    $g->is_pseudo_graph
+
+Return true if the graph has any multiedges or any self-loops,
+false otherwise.
+
+=item is_multi_graph
+
+    $g->is_multi_graph
+
+Return true if the graph has any multiedges but no self-loops,
+false otherwise.
+
+=item is_directed_acyclic_graph
+
+=item is_dag
+
+    $g->is_directed_acyclic_graph
+    $g->is_dag
+
+Return true if the graph is directed and acyclic, false otherwise.
+
+=item is_cyclic
+
+    $g->is_cyclic
+
+Return true if the graph is cyclic (contains at least one cycle).
+(This is identical to C<has_a_cycle>.)
+
+To find at least that one cycle, see L</find_a_cycle>.
+
+=item is_acyclic
+
+Return true if the graph is acyclic (does not contain any cycles).
+
+=back
+
+To find a cycle, use L<find_a_cycle>.
+
+=head2 Transitivity
+
+=over 4
+
+=item is_transitive
+
+    $g->is_transitive
+
+Return true if the graph is transitive, false otherwise.
+
+=item TransitiveClosure_Floyd_Warshall
+
+=item transitive_closure
+
+    $tcg = $g->TransitiveClosure_Floyd_Warshall
+
+Return the transitive closure graph of the graph.
+
+=back
+
+You can query the reachability from $u to $v with
+
+=over 4
+
+=item is_reachable
+
+    $tcg->is_reachable($u, $v)
+
+=back
+
+See L<Graph::TransitiveClosure> for more information about creating
+and querying transitive closures.
+
+With
+
+=over 4
+
+=item transitive_closure_matrix
+
+   $tcm = $g->transitive_closure_matrix;
+
+=back
+
+you can (create if not existing and) query the transitive closure
+matrix that underlies the transitive closure graph.  See
+L<Graph::TransitiveClosure::Matrix> for more information.
+
+=head2 Mutators
+
+=over 4
+
+=item add_vertices
+
+    $g->add_vertices('d', 'e', 'f')
+
+Add zero or more vertices to the graph.
+
+=item add_edges
+
+    $g->add_edges(['d', 'e'], ['f', 'g'])
+    $g->add_edges(qw(d e f g));
+
+Add zero or more edges to the graph.  The edges are specified as
+a list of array references, or as a list of vertices where the
+even (0th, 2nd, 4th, ...) items are start vertices and the odd
+(1st, 3rd, 5th, ...) are the corresponding end vertices.
+
+=back
+
+=head2 Accessors
+
+=over 4
+
+=item is_directed
+
+=item directed
+
+    $g->is_directed()
+    $g->directed()
+
+Return true if the graph is directed, false otherwise.
+
+=item is_undirected
+
+=item undirected
+
+    $g->is_undirected()
+    $g->undirected()
+
+Return true if the graph is undirected, false otherwise.
+
+=item is_refvertexed
+
+=item refvertexed
+
+Return true if the graph can handle references (including Perl objects)
+as vertices.
+
+=item vertices
+
+    my $V = $g->vertices
+    my @V = $g->vertices
+
+In scalar context, return the number of vertices in the graph.
+In list context, return the vertices, in no particular order.
+
+=item has_vertices
+
+    $g->has_vertices()
+
+Return true if the graph has any vertices, false otherwise.
+
+=item edges
+
+    my $E = $g->edges
+    my @E = $g->edges
+
+In scalar context, return the number of edges in the graph.
+In list context, return the edges, in no particular order.
+I<The edges are returned as anonymous arrays listing the vertices.>
+
+=item has_edges
+
+    $g->has_edges()
+
+Return true if the graph has any edges, false otherwise.
+
+=item is_connected
+
+    $g->is_connected
+
+For an undirected graph, return true is the graph is connected, false
+otherwise.  Being connected means that from every vertex it is possible
+to reach every other vertex.
+
+If the graph has been created with a true C<unionfind> parameter,
+the time complexity is (essentially) O(V), otherwise O(V log V).
+
+See also L</connected_components>, L</connected_component_by_index>,
+L</connected_component_by_vertex>, and L</same_connected_components>,
+and L</biconnectivity>.
+
+For directed graphs, see L</is_strongly_connected>
+and L</is_weakly_connected>.
+
+=item connected_components
+
+    @cc = $g->connected_components()
+
+For an undirected graph, returns the vertices of the connected
+components of the graph as a list of anonymous arrays.  The ordering
+of the anonymous arrays or the ordering of the vertices inside the
+anonymous arrays (the components) is undefined.
+
+For directed graphs, see L</strongly_connected_components>
+and L</weakly_connected_components>.
+
+=item connected_component_by_vertex
+
+    $i = $g->connected_component_by_vertex($v)
+
+For an undirected graph, return an index identifying the connected
+component the vertex belongs to, the indexing starting from zero.
+
+For the inverse, see L</connected_component_by_index>.
+
+If the graph has been created with a true C<unionfind> parameter,
+the time complexity is (essentially) O(1), otherwise O(V log V).
+
+See also L</biconnectivity>.
+
+For directed graphs, see L</strongly_connected_component_by_vertex>
+and L</weakly_connected_component_by_vertex>.
+
+=item connected_component_by_index
+
+    @v = $g->connected_component_by_index($i)
+
+For an undirected graph, return the vertices of the ith connected
+component, the indexing starting from zero.  The order of vertices is
+undefined, while the order of the connected components is same as from
+connected_components().
+
+For the inverse, see L</connected_component_by_vertex>.
+
+For directed graphs, see L</strongly_connected_component_by_index>
+and L</weakly_connected_component_by_index>.
+
+=item same_connected_components
+
+    $g->same_connected_components($u, $v, ...)
+
+For an undirected graph, return true if the vertices are in the same
+connected component.
+
+If the graph has been created with a true C<unionfind> parameter,
+the time complexity is (essentially) O(1), otherwise O(V log V).
+
+For directed graphs, see L</same_strongly_connected_components>
+and L</same_weakly_connected_components>.
+
+=item connected_graph
+
+    $cg = $g->connected_graph
+
+For an undirected graph, return its connected graph.
+
+=item connectivity_clear_cache
+
+    $g->connectivity_clear_cache
+
+See L</"Clearing cached results">.
+
+See L</"Connected Graphs and Their Components"> for further discussion.
+
+=item biconnectivity
+
+    my ($ap, $bc, $br) = $g->biconnectivity
+
+For an undirected graph, return the various biconnectivity components
+of the graph: the articulation points (cut vertices), biconnected
+components, and bridges.
+
+Note: currently only handles connected graphs.
+
+=item is_biconnected
+
+   $g->is_biconnected
+
+For an undirected graph, return true if the graph is biconnected
+(if it has no articulation points, also known as cut vertices).
+
+=item is_edge_connected
+
+   $g->is_edge_connected
+
+For an undirected graph, return true if the graph is edge-connected
+(if it has no bridges).
+
+=item is_edge_separable
+
+   $g->is_edge_separable
+
+For an undirected graph, return true if the graph is edge-separable
+(if it has bridges).
+
+=item articulation_points
+
+=item cut_vertices
+
+   $g->articulation_points
+
+For an undirected graph, return the articulation points (cut vertices)
+of the graph as a list of vertices.  The order is undefined.
+
+=item biconnected_components
+
+   $g->biconnected_components
+
+For an undirected graph, return the biconnected components of the
+graph as a list of anonymous arrays of vertices in the components.
+The ordering of the anonymous arrays or the ordering of the vertices
+inside the anonymous arrays (the components) is undefined.  Also note
+that one vertex can belong to more than one biconnected component.
+
+=item biconnected_component_by_vertex
+
+   $i = $g->biconnected_component_by_index($v)
+
+For an undirected graph, return an index identifying the biconnected
+component the vertex belongs to, the indexing starting from zero.
+
+For the inverse, see L</connected_component_by_index>.
+
+For directed graphs, see L</strongly_connected_component_by_index>
+and L</weakly_connected_component_by_index>.
+
+=item biconnected_component_by_index
+
+   @v = $g->biconnected_component_by_index($i)
+
+For an undirected graph, return the vertices in the ith biconnected
+component of the graph as an anonymous arrays of vertices in the
+component.  The ordering of the vertices within a component is
+undefined.  Also note that one vertex can belong to more than one
+biconnected component.
+
+=item same_biconnected_components
+
+    $g->same_biconnected_components($u, $v, ...)
+
+For an undirected graph, return true if the vertices are in the same
+biconnected component.
+
+=item biconnected_graph
+
+    $bcg = $g->biconnected_graph
+
+For an undirected graph, return its biconnected graph.
+
+See L</"Connected Graphs and Their Components"> for further discussion.
+
+=item bridges
+
+   $g->bridges
+
+For an undirected graph, return the bridges of the graph as a list of
+anonymous arrays of vertices in the bridges.  The order of bridges and
+the order of vertices in them is undefined.
+
+=item biconnectivity_clear_cache
+
+    $g->biconnectivity_clear_cache
+
+See L</"Clearing cached results">.
+
+=item strongly_connected
+
+=item is_strongly_connected
+
+    $g->is_strongly_connected
+
+For a directed graph, return true is the directed graph is strongly
+connected, false if not.
+
+See also L</is_weakly_connected>.
+
+For undirected graphs, see L</is_connected>, or L</is_biconnected>.
+
+=item strongly_connected_component_by_vertex
+
+    $i = $g->strongly_connected_component_by_vertex($v)
+
+For a directed graph, return an index identifying the strongly
+connected component the vertex belongs to, the indexing starting from
+zero.
+
+For the inverse, see L</strongly_connected_component_by_index>.
+
+See also L</weakly_connected_component_by_vertex>.
+
+For undirected graphs, see L</connected_components> or
+L</biconnected_components>.
+
+=item strongly_connected_component_by_index
+
+    @v = $g->strongly_connected_component_by_index($i)
+
+For a directed graph, return the vertices of the ith connected
+component, the indexing starting from zero.  The order of vertices
+within a component is undefined, while the order of the connected
+components is the as from strongly_connected_components().
+
+For the inverse, see L</strongly_connected_component_by_vertex>.
+
+For undirected graphs, see L</weakly_connected_component_by_index>.
+
+=item same_strongly_connected_components
+
+    $g->same_strongly_connected_components($u, $v, ...)
+
+For a directed graph, return true if the vertices are in the same
+strongly connected component.
+
+See also L</same_weakly_connected_components>.
+
+For undirected graphs, see L</same_connected_components> or
+L</same_biconnected_components>.
+
+=item strong_connectivity_clear_cache
+
+    $g->strong_connectivity_clear_cache
+
+See L</"Clearing cached results">.
+
+=item weakly_connected
+
+=item is_weakly_connected
+
+    $g->is_weakly_connected
+
+For a directed graph, return true is the directed graph is weakly
+connected, false if not.
+
+Weakly connected graph is also known as I<semiconnected> graph.
+
+See also L</is_strongly_connected>.
+
+For undirected graphs, see L</is_connected> or L</is_biconnected>.
+
+=item weakly_connected_components
+
+    @wcc = $g->weakly_connected_components()
+
+For a directed graph, returns the vertices of the weakly connected
+components of the graph as a list of anonymous arrays.  The ordering
+of the anonymous arrays or the ordering of the vertices inside the
+anonymous arrays (the components) is undefined.
+
+See also L</strongly_connected_components>.
+
+For undirected graphs, see L</connected_components> or
+L</biconnected_components>.
+
+=item weakly_connected_component_by_vertex
+
+    $i = $g->weakly_connected_component_by_vertex($v)
+
+For a directed graph, return an index identifying the weakly connected
+component the vertex belongs to, the indexing starting from zero.
+
+For the inverse, see L</weakly_connected_component_by_index>.
+
+For undirected graphs, see L</connected_component_by_vertex>
+and L</biconnected_component_by_vertex>.
+
+=item weakly_connected_component_by_index
+
+    @v = $g->weakly_connected_component_by_index($i)
+
+For a directed graph, return the vertices of the ith weakly connected
+component, the indexing starting zero.  The order of vertices within
+a component is undefined, while the order of the weakly connected
+components is same as from weakly_connected_components().
+
+For the inverse, see L</weakly_connected_component_by_vertex>.
+
+For undirected graphs, see L<connected_component_by_index>
+and L<biconnected_component_by_index>.
+
+=item same_weakly_connected_components
+
+    $g->same_weakly_connected_components($u, $v, ...)
+
+Return true if the vertices are in the same weakly connected component.
+
+=item weakly_connected_graph
+
+    $wcg = $g->weakly_connected_graph
+
+For a directed graph, return its weakly connected graph.
+
+For undirected graphs, see L</connected_graph> and L</biconnected_graph>.
+
+=item strongly_connected_components
+
+   my @scc = $g->strongly_connected_components;
+
+For a directed graph, return the strongly connected components as a
+list of anonymous arrays.  The elements in the anonymous arrays are
+the vertices belonging to the strongly connected component; both the
+elements and the components are in no particular order.
+
+See also L</weakly_connected_components>.
+
+For undirected graphs, see L</connected_components>,
+or see L</biconnected_components>.
+
+=item strongly_connected_graph
+
+   my $scg = $g->strongly_connected_graph;
+
+See L</"Connected Graphs and Their Components"> for further discussion.
+
+Strongly connected graphs are also known as I<kernel graphs>.
+
+See also L</weakly_connected_graph>.
+
+For undirected graphs, see L</connected_graph>, or L</biconnected_graph>.
+
+=item is_sink_vertex
+
+    $g->is_sink_vertex($v)
+
+Return true if the vertex $v is a sink vertex, false if not.  A sink
+vertex is defined as a vertex with predecessors but no successors:
+this definition means that isolated vertices are not sink vertices.
+If you want also isolated vertices, use is_successorless_vertex().
+
+=item is_source_vertex
+
+    $g->is_source_vertex($v)
+
+Return true if the vertex $v is a source vertex, false if not.  A source
+vertex is defined as a vertex with successors but no predecessors:
+the definition means that isolated vertices are not source vertices.
+If you want also isolated vertices, use is_predecessorless_vertex().
+
+=item is_successorless_vertex
+
+    $g->is_successorless_vertex($v)
+
+Return true if the vertex $v has no succcessors (no edges
+leaving the vertex), false if it has.
+
+Isolated vertices will return true: if you do not want this,
+use is_sink_vertex().
+
+=item is_successorful_vertex
+
+    $g->is_successorful_vertex($v)
+
+Return true if the vertex $v has successors, false if not.
+
+=item is_predecessorless_vertex
+
+    $g->is_predecessorless_vertex($v)
+
+Return true if the vertex $v has no predecessors (no edges
+entering the vertex), false if it has.
+
+Isolated vertices will return true: if you do not want this,
+use is_source_vertex().
+
+=item is_predecessorful_vertex
+
+    $g->is_predecessorful_vertex($v)
+
+Return true if the vertex $v has predecessors, false if not.
+
+=item is_isolated_vertex
+
+    $g->is_isolated_vertex($v)
+
+Return true if the vertex $v is an isolated vertex: no successors
+and no predecessors.
+
+=item is_interior_vertex
+
+    $g->is_interior_vertex($v)
+
+Return true if the vertex $v is an interior vertex: both successors
+and predecessors.
+
+=item is_exterior_vertex
+
+    $g->is_exterior_vertex($v)
+
+Return true if the vertex $v is an exterior vertex: has either no
+successors or no predecessors, or neither.
+
+=item is_self_loop_vertex
+
+    $g->is_self_loop_vertex($v)
+
+Return true if the vertex $v is a self loop vertex: has an edge
+from itself to itself.
+
+=item sink_vertices
+
+    @v = $g->sink_vertices()
+
+Return the sink vertices of the graph.
+In scalar context return the number of sink vertices.
+See L</is_sink_vertex> for the definition of a sink vertex.
+
+=item source_vertices
+
+    @v = $g->source_vertices()
+
+Return the source vertices of the graph.
+In scalar context return the number of source vertices.
+See L</is_source_vertex> for the definition of a source vertex.
+
+=item successorful_vertices
+
+    @v = $g->successorful_vertices()
+
+Return the successorful vertices of the graph.
+In scalar context return the number of successorful vertices.
+
+=item successorless_vertices
+
+    @v = $g->successorless_vertices()
+
+Return the successorless vertices of the graph.
+In scalar context return the number of successorless vertices.
+
+=item successors
+
+    @s = $g->successors($v)
+
+Return the immediate successor vertices of the vertex.
+
+=item neighbors
+
+=item neighbours
+
+Return the neighbo(u)ring vertices.  Also known as the I<adjacent vertices>.
+
+=item predecessorful_vertices
+
+    @v = $g->predecessorful_vertices()
+
+Return the predecessorful vertices of the graph.
+In scalar context return the number of predecessorful vertices.
+
+=item predecessorless_vertices
+
+    @v = $g->predecessorless_vertices()
+
+Return the predecessorless vertices of the graph.
+In scalar context return the number of predecessorless vertices.
+
+=item predecessors
+
+    @s = $g->predecessors($v)
+
+Return the immediate predecessor vertices of the vertex.
+
+=item isolated_vertices
+
+    @v = $g->isolated_vertices()
+
+Return the isolated vertices of the graph.
+In scalar context return the number of isolated vertices.
+See L</is_isolated_vertex> for the definition of an isolated vertex.
+
+=item interior_vertices
+
+    @v = $g->interior_vertices()
+
+Return the interior vertices of the graph.
+In scalar context return the number of interior vertices.
+See L</is_interior_vertex> for the definition of an interior vertex.
+
+=item exterior_vertices
+
+    @v = $g->exterior_vertices()
+
+Return the exterior vertices of the graph.
+In scalar context return the number of exterior vertices.
+See L</is_exterior_vertex> for the definition of an exterior vertex.
+
+=item self_loop_vertices
+
+    @v = $g->self_loop_vertices()
+
+Return the self-loop vertices of the graph.
+In scalar context return the number of self-loop vertices.
+See L</is_self_loop_vertex> for the definition of a self-loop vertex.
+
+=back
+
+=head2 Connected Graphs and Their Components
+
+In this discussion I<connected graph> refers to any of
+I<connected graphs>, I<biconnected graphs>, and I<strongly
+connected graphs>.
+
+B<NOTE>: if the vertices of the original graph are Perl objects,
+(in other words, references, so you must be using C<refvertexed>) 
+the vertices of the I<connected graph> are NOT by default usable
+as Perl objects because they are blessed into a package with
+a rather unusable name.
+
+By default, the vertex names of the I<connected graph> are formed from
+the names of the vertices of the original graph by (alphabetically
+sorting them and) concatenating their names with C<+>.  The vertex
+attribute C<subvertices> is also used to store the list (as an array
+reference) of the original vertices.  To change the 'supercomponent'
+vertex names and the whole logic of forming these supercomponents
+use the C<super_component>) option to the method calls:
+
+  $g->connected_graph(super_component => sub { ... })
+  $g->biconnected_graph(super_component => sub { ... })
+  $g->strongly_connected_graph(super_component => sub { ... })
+
+The subroutine reference gets the 'subcomponents' (the vertices of the
+original graph) as arguments, and it is supposed to return the new
+supercomponent vertex, the "stringified" form of which is used as the
+vertex name.
+
+=head2 Degree
+
+A vertex has a degree based on the number of incoming and outgoing edges.
+This really makes sense only for directed graphs.
+
+=over 4
+
+=item degree
+
+=item vertex_degree
+
+    $d = $g->degree($v)
+    $d = $g->vertex_degree($v)
+
+For directed graphs: the in-degree minus the out-degree at the vertex.
+For undirected graphs: the number of edges at the vertex.
+
+=item in_degree
+
+    $d = $g->in_degree($v)
+
+The number of incoming edges at the vertex.
+
+=item out_degree
+
+    $o = $g->out_degree($v)
+
+The number of outgoing edges at the vertex.
+
+=item average_degree
+
+   my $ad = $g->average_degree;
+
+Return the average degree taken over all vertices.
+
+=back
+
+Related methods are
+
+=over 4
+
+=item edges_at
+
+    @e = $g->edges_at($v)
+
+The union of edges from and edges to at the vertex.
+
+=item edges_from
+
+    @e = $g->edges_from($v)
+
+The edges leaving the vertex.
+
+=item edges_to
+
+    @e = $g->edges_to($v)
+
+The edges entering the vertex.
+
+=back
+
+See also L</average_degree>.
+
+=head2 Counted Vertices
+
+I<Counted vertices> are vertices with more than one instance, normally
+adding vertices is idempotent.  To enable counted vertices on a graph,
+give the C<countvertexed> parameter a true value
+
+    use Graph;
+    my $g = Graph->new(countvertexed => 1);
+
+To find out how many times the vertex has been added:
+
+=over 4
+
+=item get_vertex_count
+
+    my $c = $g->get_vertex_count($v);
+
+Return the count of the vertex, or undef if the vertex does not exist.
+
+=back
+
+=head2 Multiedges, Multivertices, Multigraphs
+
+I<Multiedges> are edges with more than one "life", meaning that one
+has to delete them as many times as they have been added.  Normally
+adding edges is idempotent (in other words, adding edges more than
+once makes no difference).
+
+There are two kinds or degrees of creating multiedges and multivertices.
+The two kinds are mutually exclusive.
+
+The weaker kind is called I<counted>, in which the edge or vertex has
+a count on it: add operations increase the count, and delete
+operations decrease the count, and once the count goes to zero, the
+edge or vertex is deleted.  If there are attributes, they all are
+attached to the same vertex.  You can think of this as the graph
+elements being I<refcounted>, or I<reference counted>, if that sounds
+more familiar.
+
+The stronger kind is called (true) I<multi>, in which the edge or vertex
+really has multiple separate identities, so that you can for example
+attach different attributes to different instances.
+
+To enable multiedges on a graph:
+
+    use Graph;
+    my $g0 = Graph->new(countedged => 1);
+    my $g0 = Graph->new(multiedged => 1);
+
+Similarly for vertices
+
+    use Graph;
+    my $g1 = Graph->new(countvertexed => 1);
+    my $g1 = Graph->new(multivertexed => 1);
+
+You can test for these by
+
+=over 4
+
+=item is_countedged
+
+=item countedged
+
+    $g->is_countedged
+    $g->countedged
+
+Return true if the graph is countedged.
+
+=item is_countvertexed
+
+=item countvertexed
+
+    $g->is_countvertexed
+    $g->countvertexed
+
+Return true if the graph is countvertexed.
+
+=item is_multiedged
+
+=item multiedged
+
+    $g->is_multiedged
+    $g->multiedged
+
+Return true if the graph is multiedged.
+
+=item is_multivertexed
+
+=item multivertexed
+
+    $g->is_multivertexed
+    $g->multivertexed
+
+Return true if the graph is multivertexed.
+
+=back
+
+A multiedged (either the weak kind or the strong kind) graph is
+a I<multigraph>, for which you can test with C<is_multi_graph()>.
+
+B<NOTE>: The various graph algorithms do not in general work well with
+multigraphs (they often assume I<simple graphs>, that is, no
+multiedges or loops), and no effort has been made to test the
+algorithms with multigraphs.
+
+vertices() and edges() will return the multiple elements: if you want
+just the unique elements, use
+
+=over 4
+
+=item unique_vertices
+
+=item unique_edges
+
+    @uv = $g->unique_vertices; # unique
+    @mv = $g->vertices;        # possible multiples
+    @ue = $g->unique_edges;
+    @me = $g->edges;
+
+=back
+
+If you are using (the stronger kind of) multielements, you should use
+the I<by_id> variants:
+
+=over 4
+
+=item add_vertex_by_id
+
+=item has_vertex_by_id
+
+=item delete_vertex_by_id
+
+=item add_edge_by_id
+
+=item has_edge_by_id
+
+=item delete_edge_by_id
+
+=back
+
+    $g->add_vertex_by_id($v, $id)
+    $g->has_vertex_by_id($v, $id)
+    $g->delete_vertex_by_id($v, $id)
+
+    $g->add_edge_by_id($u, $v, $id)
+    $g->has_edge_by_id($u, $v, $id)
+    $g->delete_edge_by_id($u, $v, $id)
+
+When you delete the last vertex/edge in a multivertex/edge, the whole
+vertex/edge is deleted.  You can use add_vertex()/add_edge() on
+a multivertex/multiedge graph, in which case an id is generated
+automatically.  To find out which the generated id was, you need
+to use
+
+=over 4
+
+=item add_vertex_get_id
+
+=item add_edge_get_id
+
+=back
+
+    $idv = $g->add_vertex_get_id($v)
+    $ide = $g->add_edge_get_id($u, $v)
+
+To return all the ids of vertices/edges in a multivertex/multiedge, use
+
+=over 4
+
+=item get_multivertex_ids
+
+=item get_multiedge_ids
+
+=back
+
+    $g->get_multivertex_ids($v)
+    $g->get_multiedge_ids($u, $v)
+
+The ids are returned in random order.
+
+To find out how many times the edge has been added (this works for
+either kind of multiedges):
+
+=over 4
+
+=item get_edge_count
+
+    my $c = $g->get_edge_count($u, $v);
+
+Return the count (the "countedness") of the edge, or undef if the
+edge does not exist.
+
+=back
+
+The following multi-entity utility functions exist, mirroring
+the non-multi vertices and edges:
+
+=over 4
+
+=item add_weighted_edge_by_id
+
+=item add_weighted_edges_by_id
+
+=item add_weighted_path_by_id
+
+=item add_weighted_vertex_by_id
+
+=item add_weighted_vertices_by_id
+
+=item delete_edge_weight_by_id
+
+=item delete_vertex_weight_by_id
+
+=item get_edge_weight_by_id
+
+=item get_vertex_weight_by_id
+
+=item has_edge_weight_by_id
+
+=item has_vertex_weight_by_id
+
+=item set_edge_weight_by_id
+
+=item set_vertex_weight_by_id
+
+=back
+
+=head2 Topological Sort
+
+=over 4
+
+=item topological_sort
+
+=item toposort
+
+    my @ts = $g->topological_sort;
+
+Return the vertices of the graph sorted topologically.  Note that
+there may be several possible topological orderings; one of them
+is returned.
+
+If the graph contains a cycle, a fatal error is thrown, you
+can either use C<eval> to trap that, or supply the C<empty_if_cyclic>
+argument with a true value
+
+    my @ts = $g->topological_sort(empty_if_cyclic => 1);
+
+in which case an empty array is returned if the graph is cyclic.
+
+=back
+
+=head2 Minimum Spanning Trees (MST)
+
+Minimum Spanning Trees or MSTs are tree subgraphs derived from an
+undirected graph.  MSTs "span the graph" (covering all the vertices)
+using as lightly weighted (hence the "minimum") edges as possible.
+
+=over 4
+
+=item MST_Kruskal
+
+    $mstg = $g->MST_Kruskal;
+
+Returns the Kruskal MST of the graph.
+
+=item MST_Prim
+
+    $mstg = $g->MST_Prim(%opt);
+
+Returns the Prim MST of the graph.
+
+You can choose the first vertex with $opt{ first_root }.
+
+=item MST_Dijkstra
+
+=item minimum_spanning_tree
+
+    $mstg = $g->MST_Dijkstra;
+    $mstg = $g->minimum_spanning_tree;
+
+Aliases for MST_Prim.
+
+=back
+
+=head2 Single-Source Shortest Paths (SSSP)
+
+Single-source shortest paths, also known as Shortest Path Trees
+(SPTs).  For either a directed or an undirected graph, return a (tree)
+subgraph that from a single start vertex (the "single source") travels
+the shortest possible paths (the paths with the lightest weights) to
+all the other vertices.  Note that the SSSP is neither reflexive (the
+shortest paths do not include the zero-length path from the source
+vertex to the source vertex) nor transitive (the shortest paths do not
+include transitive closure paths).  If no weight is defined for an
+edge, 1 (one) is assumed.
+
+=over 4
+
+=item SPT_Dijkstra
+
+    $sptg = $g->SPT_Dijkstra($root)
+    $sptg = $g->SPT_Dijkstra(%opt)
+
+Return as a graph the the single-source shortest paths of the graph
+using Dijkstra's algorithm.  The graph cannot contain negative edges
+(negative edges cause the algorithm to abort with an error message
+C<Graph::SPT_Dijkstra: edge ... is negative>).
+
+You can choose the first vertex of the result with either a single
+vertex argument or with $opt{ first_root }, otherwise a random vertex
+is chosen.
+
+B<NOTE>: note that all the vertices might not be reachable from the
+selected (explicit or random) start vertex.
+
+The start vertex is be available as the graph attribute
+C<SPT_Dijkstra_root>).
+
+The result weights of vertices can be retrieved from the result graph by
+
+	my $w = $sptg->get_vertex_attribute($v, 'weight');
+
+The predecessor vertex of a vertex in the result graph
+can be retrieved by
+
+	my $u = $sptg->get_vertex_attribute($v, 'p');
+
+("A successor vertex" cannot be retrieved as simply because a single
+vertex can have several successors.  You can first find the
+C<neighbors()> vertices and then remove the predecessor vertex.)
+
+If you want to find the shortest path between two vertices,
+see L</SP_Dijkstra>.
+
+=item SSSP_Dijkstra
+
+=item single_source_shortest_paths
+
+Aliases for SPT_Dijkstra.
+
+=item SP_Dijkstra
+
+    @path = $g->SP_Dijkstra($u, $v)
+
+Return the vertices in the shortest path in the graph $g between the
+two vertices $u, $v.  If no path can be found, an empty list is returned.
+
+Uses SPT_Dijkstra().
+
+=item SPT_Dijkstra_clear_cache
+
+    $g->SPT_Dijkstra_clear_cache
+
+See L</"Clearing cached results">.
+
+=item SPT_Bellman_Ford
+
+    $sptg = $g->SPT_Bellman_Ford(%opt)
+
+Return as a graph the single-source shortest paths of the graph using
+Bellman-Ford's algorithm.  The graph can contain negative edges but
+not negative cycles (negative cycles cause the algorithm to abort
+with an error message C<Graph::SPT_Bellman_Ford: negative cycle exists/>).
+
+You can choose the start vertex of the result with either a single
+vertex argument or with $opt{ first_root }, otherwise a random vertex
+is chosen.
+
+B<NOTE>: note that all the vertices might not be reachable from the
+selected (explicit or random) start vertex.
+
+The start vertex is be available as the graph attribute
+C<SPT_Bellman_Ford_root>).
+
+The result weights of vertices can be retrieved from the result graph by
+
+	my $w = $sptg->get_vertex_attribute($v, 'weight');
+
+The predecessor vertex of a vertex in the result graph
+can be retrieved by
+
+	my $u = $sptg->get_vertex_attribute($v, 'p');
+
+("A successor vertex" cannot be retrieved as simply because a single
+vertex can have several successors.  You can first find the
+C<neighbors()> vertices and then remove the predecessor vertex.)
+
+If you want to find the shortes path between two vertices,
+see L</SP_Bellman_Ford>.
+
+=item SSSP_Bellman_Ford
+
+Alias for SPT_Bellman_Ford.
+
+=item SP_Bellman_Ford
+
+    @path = $g->SP_Bellman_Ford($u, $v)
+
+Return the vertices in the shortest path in the graph $g between the
+two vertices $u, $v.  If no path can be found, an empty list is returned.
+
+Uses SPT_Bellman_Ford().
+
+=item SPT_Bellman_Ford_clear_cache
+
+    $g->SPT_Bellman_Ford_clear_cache
+
+See L</"Clearing cached results">.
+
+=back
+
+=head2 All-Pairs Shortest Paths (APSP)
+
+For either a directed or an undirected graph, return the APSP object
+describing all the possible paths between any two vertices of the
+graph.  If no weight is defined for an edge, 1 (one) is assumed.
+
+=over 4
+
+=item APSP_Floyd_Warshall
+
+=item all_pairs_shortest_paths
+
+    my $apsp = $g->APSP_Floyd_Warshall(...);
+
+Return the all-pairs shortest path object computed from the graph
+using Floyd-Warshall's algorithm.  The length of a path between two
+vertices is the sum of weight attribute of the edges along the
+shortest path between the two vertices.  If no weight attribute name
+is specified explicitly
+
+    $g->APSP_Floyd_Warshall(attribute_name => 'height');
+
+the attribute C<weight> is assumed.
+
+B<If an edge has no defined weight attribute, the value of one is
+assumed when getting the attribute.>
+
+Once computed, you can query the APSP object with
+
+=over 8
+
+=item path_length
+
+    my $l = $apsp->path_length($u, $v);
+
+Return the length of the shortest path between the two vertices.
+
+=item path_vertices
+
+    my @v = $apsp->path_vertices($u, $v);
+
+Return the list of vertices along the shortest path.
+
+=item path_predecessor
+
+   my $u = $apsp->path_predecessor($v);
+
+Returns the predecessor of vertex $v in the all-pairs shortest paths.
+
+=back
+
+=over 8
+
+=item average_path_length
+
+    my $apl = $g->average_path_length; # All vertex pairs.
+
+    my $apl = $g->average_path_length($u); # From $u.
+    my $apl = $g->average_path_length($u, undef); # From $u.
+
+    my $apl = $g->average_path_length($u, $v); # From $u to $v.
+
+    my $apl = $g->average_path_length(undef, $v); # To $v.
+
+Return the average (shortest) path length over all the vertex pairs of
+the graph, from a vertex, between two vertices, and to a vertex.
+
+=item longest_path
+
+    my @lp = $g->longest_path;
+    my $lp = $g->longest_path;
+
+In scalar context return the I<longest shortest> path length over all
+the vertex pairs of the graph.  In list context return the vertices
+along a I<longest shortest> path.  Note that there might be more than
+one such path; this interfaces return a random one of them.
+
+=item diameter
+
+=item graph_diameter
+
+    my $gd = $g->diameter;
+
+The longest path over all the vertex pairs is known as the
+I<graph diameter>.
+
+=item shortest_path
+
+    my @sp = $g->shortest_path;
+    my $sp = $g->shortest_path;
+
+In scalar context return the shortest length over all the vertex pairs
+of the graph.  In list context return the vertices along a shortest
+path.  Note that there might be more than one such path; this
+interface returns a random one of them.
+
+=item radius
+
+    my $gr = $g->radius;
+
+The I<shortest longest> path over all the vertex pairs is known as the
+I<graph radius>.  See also L</diameter>.
+
+=item center_vertices
+
+=item centre_vertices
+
+    my @c = $g->center_vertices;
+    my @c = $g->center_vertices($delta);
+
+The I<graph center> is the set of vertices for which the I<vertex
+eccentricity> is equal to the I<graph radius>.  The vertices are
+returned in random order.  By specifying a delta value you can widen
+the criterion from strict equality (handy for non-integer edge weights).
+
+=item vertex_eccentricity
+
+    my $ve = $g->vertex_eccentricity($v);
+
+The longest path to a vertex is known as the I<vertex eccentricity>.
+If the graph is unconnected, returns Inf.
+
+=back
+
+You can walk through the matrix of the shortest paths by using
+
+=over 4
+
+=item for_shortest_paths
+
+    $n = $g->for_shortest_paths($callback)
+
+The number of shortest paths is returned (this should be equal to V*V).
+The $callback is a sub reference that receives four arguments:
+the transitive closure object from Graph::TransitiveClosure, the two
+vertices, and the index to the current shortest paths (0..V*V-1).
+
+=back
+
+=back
+
+=head2 Clearing cached results
+
+For many graph algorithms there are several different but equally valid
+results.  (Pseudo)Randomness is used internally by the Graph module to
+for example pick a random starting vertex, and to select random edges
+from a vertex.
+
+For efficiency the computed result is often cached to avoid
+recomputing the potentially expensive operation, and this also gives
+additional determinism (once a correct result has been computed, the
+same result will always be given).
+
+However, sometimes the exact opposite is desireable, and the possible
+alternative results are wanted (within the limits of the pseudorandomness:
+not all the possible solutions are guaranteed to be returned, usually only
+a subset is retuned).  To undo the caching, the following methods are
+available:
+
+=over 4
+
+=item *
+
+connectivity_clear_cache
+
+Affects L</connected_components>, L</connected_component_by_vertex>,
+L</connected_component_by_index>, L</same_connected_components>,
+L</connected_graph>, L</is_connected>, L</is_weakly_connected>,
+L</weakly_connected_components>, L</weakly_connected_component_by_vertex>,
+L</weakly_connected_component_by_index>, L</same_weakly_connected_components>,
+L</weakly_connected_graph>.
+
+=item *
+
+biconnectivity_clear_cache
+
+Affects L</biconnected_components>,
+L</biconnected_component_by_vertex>,
+L</biconnected_component_by_index>, L</is_edge_connected>,
+L</is_edge_separable>, L</articulation_points>, L</cut_vertices>,
+L</is_biconnected>, L</biconnected_graph>,
+L</same_biconnected_components>, L</bridges>.
+
+=item *
+
+strong_connectivity_clear_cache
+
+Affects L</strongly_connected_components>,
+L</strongly_connected_component_by_vertex>,
+L</strongly_connected_component_by_index>,
+L</same_strongly_connected_components>, L</is_strongly_connected>,
+L</strongly_connected>, L</strongly_connected_graph>.
+
+=item *
+
+SPT_Dijkstra_clear_cache
+
+Affects L</SPT_Dijkstra>, L</SSSP_Dijkstra>, L</single_source_shortest_paths>,
+L</SP_Dijkstra>.
+
+=item *
+
+SPT_Bellman_Ford_clear_cache
+
+Affects L</SPT_Bellman_Ford>, L</SSSP_Bellman_Ford>, L</SP_Bellman_Ford>.
+
+=back
+
+Note that any such computed and cached results are of course always
+automatically discarded whenever the graph is modified.
+
+=head2 Random
+
+You can either ask for random elements of existing graphs or create
+random graphs.
+
+=over 4
+
+=item random_vertex
+
+    my $v = $g->random_vertex;
+
+Return a random vertex of the graph, or undef if there are no vertices.
+
+=item random_edge
+
+    my $e = $g->random_edge;
+
+Return a random edge of the graph as an array reference having the
+vertices as elements, or undef if there are no edges.
+
+=item random_successor
+
+    my $v = $g->random_successor($v);
+
+Return a random successor of the vertex in the graph, or undef if there
+are no successors.
+
+=item random_predecessor
+
+    my $u = $g->random_predecessor($v);
+
+Return a random predecessor of the vertex in the graph, or undef if there
+are no predecessors.
+
+=item random_graph
+
+    my $g = Graph->random_graph(%opt);
+
+Construct a random graph.  The I<%opt> B<must> contain the C<vertices>
+argument
+
+    vertices => vertices_def
+
+where the I<vertices_def> is one of
+
+=over 8
+
+=item *
+
+an array reference where the elements of the array reference are the
+vertices
+
+=item *
+
+a number N in which case the vertices will be integers 0..N-1
+
+=back
+
+=back
+
+The %opt may have either of the argument C<edges> or the argument
+C<edges_fill>.  Both are used to define how many random edges to
+add to the graph; C<edges> is an absolute number, while C<edges_fill>
+is a relative number (relative to the number of edges in a complete
+graph, C).  The number of edges can be larger than C, but only if the
+graph is countedged.  The random edges will not include self-loops.
+If neither C<edges> nor C<edges_fill> is specified, an C<edges_fill>
+of 0.5 is assumed.
+
+If you want repeatable randomness (what is an oxymoron?)
+you can use the C<random_seed> option:
+
+    $g = Graph->random_graph(vertices => 10, random_seed => 1234);
+
+As this uses the standard Perl srand(), the usual caveat applies:
+use it sparingly, and consider instead using a single srand() call
+at the top level of your application.
+
+The default random distribution of edges is flat, that is, any pair of
+vertices is equally likely to appear.  To define your own distribution,
+use the C<random_edge> option:
+
+    $g = Graph->random_graph(vertices => 10, random_edge => \&d);
+
+where C<d> is a code reference receiving I<($g, $u, $v, $p)> as
+parameters, where the I<$g> is the random graph, I<$u> and I<$v> are
+the vertices, and the I<$p> is the probability ([0,1]) for a flat
+distribution.  It must return a probability ([0,1]) that the vertices
+I<$u> and I<$v> have an edge between them.  Note that returning one
+for a particular pair of vertices doesn't guarantee that the edge will
+be present in the resulting graph because the required number of edges
+might be reached before that particular pair is tested for the
+possibility of an edge.  Be very careful to adjust also C<edges>
+or C<edges_fill> so that there is a possibility of the filling process
+terminating.
+
+=head2 Attributes
+
+You can attach free-form attributes (key-value pairs, in effect a full
+Perl hash) to each vertex, edge, and the graph itself.
+
+Note that attaching attributes does slow down some other operations
+on the graph by a factor of three to ten.  For example adding edge
+attributes does slow down anything that walks through all the edges.
+
+For vertex attributes:
+
+=over 4
+
+=item set_vertex_attribute
+
+    $g->set_vertex_attribute($v, $name, $value)
+
+Set the named vertex attribute.
+
+If the vertex does not exist, the set_...() will create it, and the
+other vertex attribute methods will return false or empty.
+
+B<NOTE: any attributes beginning with an underscore/underline (_)
+are reserved for the internal use of the Graph module.>
+
+=item get_vertex_attribute
+
+    $value = $g->get_vertex_attribute($v, $name)
+
+Return the named vertex attribute.
+
+=item has_vertex_attribute
+
+    $g->has_vertex_attribute($v, $name)
+
+Return true if the vertex has an attribute, false if not.
+
+=item delete_vertex_attribute
+
+    $g->delete_vertex_attribute($v, $name)
+
+Delete the named vertex attribute.
+
+=item set_vertex_attributes
+
+    $g->set_vertex_attributes($v, $attr)
+
+Set all the attributes of the vertex from the anonymous hash $attr.
+
+B<NOTE>: any attributes beginning with an underscore (C<_>) are
+reserved for the internal use of the Graph module.
+
+=item get_vertex_attributes
+
+    $attr = $g->get_vertex_attributes($v)
+
+Return all the attributes of the vertex as an anonymous hash.
+
+=item get_vertex_attribute_names
+
+    @name = $g->get_vertex_attribute_names($v)
+
+Return the names of vertex attributes.
+
+=item get_vertex_attribute_values
+
+    @value = $g->get_vertex_attribute_values($v)
+
+Return the values of vertex attributes.
+
+=item has_vertex_attributes
+
+    $g->has_vertex_attributes($v)
+
+Return true if the vertex has any attributes, false if not.
+
+=item delete_vertex_attributes
+
+    $g->delete_vertex_attributes($v)
+
+Delete all the attributes of the named vertex.
+
+=back
+
+If you are using multivertices, use the I<by_id> variants:
+
+=over 4
+
+=item set_vertex_attribute_by_id
+
+=item get_vertex_attribute_by_id
+
+=item has_vertex_attribute_by_id
+
+=item delete_vertex_attribute_by_id
+
+=item set_vertex_attributes_by_id
+
+=item get_vertex_attributes_by_id
+
+=item get_vertex_attribute_names_by_id
+
+=item get_vertex_attribute_values_by_id
+
+=item has_vertex_attributes_by_id
+
+=item delete_vertex_attributes_by_id
+
+    $g->set_vertex_attribute_by_id($v, $id, $name, $value)
+    $g->get_vertex_attribute_by_id($v, $id, $name)
+    $g->has_vertex_attribute_by_id($v, $id, $name)
+    $g->delete_vertex_attribute_by_id($v, $id, $name)
+    $g->set_vertex_attributes_by_id($v, $id, $attr)
+    $g->get_vertex_attributes_by_id($v, $id)
+    $g->get_vertex_attribute_values_by_id($v, $id)
+    $g->get_vertex_attribute_names_by_id($v, $id)
+    $g->has_vertex_attributes_by_id($v, $id)
+    $g->delete_vertex_attributes_by_id($v, $id)
+
+=back
+
+For edge attributes:
+
+=over 4
+
+=item set_edge_attribute
+
+    $g->set_edge_attribute($u, $v, $name, $value)
+
+Set the named edge attribute.
+
+If the edge does not exist, the set_...() will create it, and the other
+edge attribute methods will return false or empty.
+
+B<NOTE>: any attributes beginning with an underscore (C<_>) are
+reserved for the internal use of the Graph module.
+
+=item get_edge_attribute
+
+    $value = $g->get_edge_attribute($u, $v, $name)
+
+Return the named edge attribute.
+
+=item has_edge_attribute
+
+    $g->has_edge_attribute($u, $v, $name)
+
+Return true if the edge has an attribute, false if not.
+
+=item delete_edge_attribute
+
+    $g->delete_edge_attribute($u, $v, $name)
+
+Delete the named edge attribute.
+
+=item set_edge_attributes
+
+    $g->set_edge_attributes($u, $v, $attr)
+
+Set all the attributes of the edge from the anonymous hash $attr.
+
+B<NOTE>: any attributes beginning with an underscore (C<_>) are
+reserved for the internal use of the Graph module.
+
+=item get_edge_attributes
+
+    $attr = $g->get_edge_attributes($u, $v)
+
+Return all the attributes of the edge as an anonymous hash.
+
+=item get_edge_attribute_names
+
+    @name = $g->get_edge_attribute_names($u, $v)
+
+Return the names of edge attributes.
+
+=item get_edge_attribute_values
+
+    @value = $g->get_edge_attribute_values($u, $v)
+
+Return the values of edge attributes.
+
+=item has_edge_attributes
+
+    $g->has_edge_attributes($u, $v)
+
+Return true if the edge has any attributes, false if not.
+
+=item delete_edge_attributes
+
+    $g->delete_edge_attributes($u, $v)
+
+Delete all the attributes of the named edge.
+
+=back
+
+If you are using multiedges, use the I<by_id> variants:
+
+=over 4
+
+=item set_edge_attribute_by_id
+
+=item get_edge_attribute_by_id
+
+=item has_edge_attribute_by_id
+
+=item delete_edge_attribute_by_id
+
+=item set_edge_attributes_by_id
+
+=item get_edge_attributes_by_id
+
+=item get_edge_attribute_names_by_id
+
+=item get_edge_attribute_values_by_id
+
+=item has_edge_attributes_by_id
+
+=item delete_edge_attributes_by_id
+
+    $g->set_edge_attribute_by_id($u, $v, $id, $name, $value)
+    $g->get_edge_attribute_by_id($u, $v, $id, $name)
+    $g->has_edge_attribute_by_id($u, $v, $id, $name)
+    $g->delete_edge_attribute_by_id($u, $v, $id, $name)
+    $g->set_edge_attributes_by_id($u, $v, $id, $attr)
+    $g->get_edge_attributes_by_id($u, $v, $id)
+    $g->get_edge_attribute_values_by_id($u, $v, $id)
+    $g->get_edge_attribute_names_by_id($u, $v, $id)
+    $g->has_edge_attributes_by_id($u, $v, $id)
+    $g->delete_edge_attributes_by_id($u, $v, $id)
+
+=back
+
+For graph attributes:
+
+=over 4
+
+=item set_graph_attribute
+
+    $g->set_graph_attribute($name, $value)
+
+Set the named graph attribute.
+
+B<NOTE>: any attributes beginning with an underscore (C<_>) are
+reserved for the internal use of the Graph module.
+
+=item get_graph_attribute
+
+    $value = $g->get_graph_attribute($name)
+
+Return the named graph attribute.
+
+=item has_graph_attribute
+
+    $g->has_graph_attribute($name)
+
+Return true if the graph has an attribute, false if not.
+
+=item delete_graph_attribute
+
+    $g->delete_graph_attribute($name)
+
+Delete the named graph attribute.
+
+=item set_graph_attributes
+
+    $g->get_graph_attributes($attr)
+
+Set all the attributes of the graph from the anonymous hash $attr.
+
+B<NOTE>: any attributes beginning with an underscore (C<_>) are
+reserved for the internal use of the Graph module.
+
+=item get_graph_attributes
+
+    $attr = $g->get_graph_attributes()
+
+Return all the attributes of the graph as an anonymous hash.
+
+=item get_graph_attribute_names
+
+    @name = $g->get_graph_attribute_names()
+
+Return the names of graph attributes.
+
+=item get_graph_attribute_values
+
+    @value = $g->get_graph_attribute_values()
+
+Return the values of graph attributes.
+
+=item has_graph_attributes
+
+    $g->has_graph_attributes()
+
+Return true if the graph has any attributes, false if not.
+
+=item delete_graph_attributes
+
+    $g->delete_graph_attributes()
+
+Delete all the attributes of the named graph.
+
+=back
+
+=head2 Weighted
+
+As convenient shortcuts the following methods add, query, and
+manipulate the attribute C<weight> with the specified value to the
+respective Graph elements.
+
+=over 4
+
+=item add_weighted_edge
+
+    $g->add_weighted_edge($u, $v, $weight)
+
+=item add_weighted_edges
+
+    $g->add_weighted_edges($u1, $v1, $weight1, ...)
+
+=item add_weighted_path
+
+    $g->add_weighted_path($v1, $weight1, $v2, $weight2, $v3, ...)
+
+=item add_weighted_vertex
+
+    $g->add_weighted_vertex($v, $weight)
+
+=item add_weighted_vertices
+
+    $g->add_weighted_vertices($v1, $weight1, $v2, $weight2, ...)
+
+=item delete_edge_weight
+
+    $g->delete_edge_weight($u, $v)
+
+=item delete_vertex_weight
+
+    $g->delete_vertex_weight($v)
+
+=item get_edge_weight
+
+    $g->get_edge_weight($u, $v)
+
+=item get_vertex_weight
+
+    $g->get_vertex_weight($v)
+
+=item has_edge_weight
+
+    $g->has_edge_weight($u, $v)
+
+=item has_vertex_weight
+
+    $g->has_vertex_weight($v)
+
+=item set_edge_weight
+
+    $g->set_edge_weight($u, $v, $weight)
+
+=item set_vertex_weight
+
+    $g->set_vertex_weight($v, $weight)
+
+=back
+
+=head2 Isomorphism
+
+Two graphs being I<isomorphic> means that they are structurally the
+same graph, the difference being that the vertices might have been
+I<renamed> or I<substituted>.  For example in the below example $g0
+and $g1 are isomorphic: the vertices C<b c d> have been renamed as
+C<z x y>.
+
+	$g0 = Graph->new;
+	$g0->add_edges(qw(a b a c c d));
+	$g1 = Graph->new;
+	$g1->add_edges(qw(a x x y a z));
+
+In the general case determining isomorphism is I<NP-hard>, in other
+words, really hard (time-consuming), no other ways of solving the problem
+are known than brute force check of of all the possibilities (with possible
+optimization tricks, of course, but brute force still rules at the end of
+the day).
+
+A B<very rough guess> at whether two graphs B<could> be isomorphic
+is possible via the method
+
+=over 4
+
+=item could_be_isomorphic
+
+    $g0->could_be_isomorphic($g1)    
+
+=back
+
+If the graphs do not have the same number of vertices and edges, false
+is returned.  If the distribution of I<in-degrees> and I<out-degrees>
+at the vertices of the graphs does not match, false is returned.
+Otherwise, true is returned.
+
+What is actually returned is the maximum number of possible isomorphic
+graphs between the two graphs, after the above sanity checks have been
+conducted.  It is basically the product of the factorials of the
+absolute values of in-degrees and out-degree pairs at each vertex,
+with the isolated vertices ignored (since they could be reshuffled and
+renamed arbitrarily).  Note that for large graphs the product of these
+factorials can overflow the maximum presentable number (the floating
+point number) in your computer (in Perl) and you might get for example
+I<Infinity> as the result.
+
+=head2 Miscellaneous
+
+The "expect" methods can be used to test a graph and croak if the
+graph is not as expected.
+
+=over 4
+
+=item expect_acyclic
+
+=item expect_dag
+
+=item expect_directed
+
+=item expect_multiedged
+
+=item expect_multivertexed
+
+=item expect_non_multiedged
+
+=item expect_non_multivertexed
+
+=item expect_undirected
+
+=back
+
+In many algorithms it is useful to have a value representing the
+infinity.  The Graph provides (and itself uses):
+
+=over 4
+
+=item Infinity
+
+(Not exported, use Graph::Infinity explicitly)
+
+=back
+
+=head2 Size Requirements
+
+A graph takes up at least 1172 bytes of memory.
+
+A vertex takes up at least 100 bytes of memory.
+
+An edge takes up at least 400 bytes of memory.
+
+(A Perl scalar value takes 16 bytes, or 12 bytes if it's a reference.)
+
+These size approximations are B<very> approximate and optimistic
+(they are based on total_size() of Devel::Size).  In real life many
+factors affect these numbers, for example how Perl is configured.
+The numbers are for a 32-bit platform and for Perl 5.8.8.
+
+Roughly, the above numbers mean that in a megabyte of memory you can
+fit for example a graph of about 1000 vertices and about 2500 edges.
+
+=head2 Hyperedges, hypervertices, hypergraphs
+
+B<BEWARE>: this is a rather thinly tested feature, and the theory
+is even less so.  Do not expect this to stay as it is (or at all)
+in future releases.
+
+B<NOTE>: most usual graph algorithms (and basic concepts) break
+horribly (or at least will look funny) with these hyperthingies.
+Caveat emptor.
+
+Hyperedges are edges that connect a number of vertices different
+from the usual two.
+
+Hypervertices are vertices that consist of a number of vertices
+different from the usual one.
+
+Note that for hypervertices there is an asymmetry: when adding
+hypervertices, the single vertices are also implicitly added.
+
+Hypergraphs are graphs with hyperedges.
+
+To enable hyperness when constructing Graphs use the C<hyperedged>
+and C<hypervertexed> attributes:
+
+   my $h = Graph->new(hyperedged => 1, hypervertexed => 1);
+
+To add hypervertexes, either explicitly use more than one vertex (or,
+indeed, I<no> vertices) when using add_vertex()
+
+   $h->add_vertex("a", "b")
+   $h->add_vertex()
+
+or implicitly with array references when using add_edge()
+
+   $h->add_edge(["a", "b"], "c")
+   $h->add_edge()
+
+Testing for existence and deletion of hypervertices and hyperedges
+works similarly.
+
+To test for hyperness of a graph use the
+
+=over 4
+
+=item is_hypervertexed
+
+=item hypervertexed
+
+    $g->is_hypervertexed
+    $g->hypervertexed
+
+=item is_hyperedged
+
+=item hyperedged
+
+    $g->is_hyperedged
+    $g->hyperedged
+
+=back
+
+Since hypervertices consist of more than one vertex:
+
+=over 4
+
+=item vertices_at
+
+    $g->vertices_at($v)
+
+=back
+
+Return the vertices at the vertex.  This may return just the vertex
+or also other vertices.
+
+To go with the concept of undirected in normal (non-hyper) graphs,
+there is a similar concept of omnidirected I<(this is my own coinage,
+"all-directions")> for hypergraphs, and you can naturally test for it by
+
+=over 4
+
+=item is_omnidirected
+
+=item omnidirected
+
+=item is_omniedged
+
+=item omniedged
+
+   $g->is_omniedged
+
+   $g->omniedged
+
+   $g->is_omnidirected
+
+   $g->omnidirected
+
+Return true if the graph is omnidirected (edges have no direction),
+false if not.
+
+=back
+
+You may be wondering why on earth did I make up this new concept, why
+didn't the "undirected" work for me?  Well, because of this:
+
+   $g = Graph->new(hypervertexed => 1, omnivertexed => 1);
+
+That's right, vertices can be omni, too - and that is indeed the
+default.  You can turn it off and then $g->add_vertex(qw(a b)) no
+more means adding also the (hyper)vertex qw(b a).  In other words,
+the "directivity" is orthogonal to (or independent of) the number of
+vertices in the vertex/edge.
+
+=over 4
+
+=item is_omnivertexed
+
+=item omnivertexed
+
+=back
+
+Another oddity that fell out of the implementation is the uniqueness
+attribute, that comes naturally in C<uniqedged> and C<uniqvertexed>
+flavours.  It does what it sounds like, to unique or not the vertices
+participating in edges and vertices (is the hypervertex qw(a b a) the
+same as the hypervertex qw(a b), for example).  Without too much
+explanation:
+
+=over 4
+
+=item is_uniqedged
+
+=item uniqedged
+
+=item is_uniqvertexed
+
+=item uniqvertexed
+
+=back
+
+=head2 Backward compatibility with Graph 0.2
+
+The Graph 0.2 (and 0.2xxxx) had the following features
+
+=over 4
+
+=item *
+
+vertices() always sorted the vertex list, which most of the time is
+unnecessary and wastes CPU.
+
+=item *
+
+edges() returned a flat list where the begin and end vertices of the
+edges were intermingled: every even index had an edge begin vertex,
+and every odd index had an edge end vertex.  This had the unfortunate
+consequence of C<scalar(@e = edges)> being twice the number of edges,
+and complicating any algorithm walking through the edges.
+
+=item *
+
+The vertex list returned by edges() was sorted, the primary key being
+the edge begin vertices, and the secondary key being the edge end vertices.
+
+=item *
+
+The attribute API was oddly position dependent and dependent
+on the number of arguments.  Use ..._graph_attribute(),
+..._vertex_attribute(), ..._edge_attribute() instead.
+
+=back
+
+B<In future releases of Graph (any release after 0.50) the 0.2xxxx
+compatibility will be removed.  Upgrade your code now.>
+
+If you want to continue using these (mis)features you can use the
+C<compat02> flag when creating a graph:
+
+    my $g = Graph->new(compat02 => 1);
+
+This will change the vertices() and edges() appropriately.  This,
+however, is not recommended, since it complicates all the code using
+vertices() and edges().  Instead it is recommended that the
+vertices02() and edges02() methods are used.  The corresponding new
+style (unsorted, and edges() returning a list of references) methods
+are called vertices05() and edges05().
+
+To test whether a graph has the compatibility turned on
+
+=over 4
+
+=item is_compat02
+
+=item compat02
+
+    $g->is_compat02
+    $g->compat02
+
+=back
+
+The following are not backward compatibility methods, strictly
+speaking, because they did not exist before.
+
+=over 4
+
+=item edges02
+
+Return the edges as a flat list of vertices, elements at even indices
+being the start vertices and elements at odd indices being the end
+vertices.
+
+=item edges05
+
+Return the edges as a list of array references, each element
+containing the vertices of each edge.  (This is not a backward
+compatibility interface as such since it did not exist before.)
+
+=item vertices02
+
+Return the vertices in sorted order.
+
+=item vertices05
+
+Return the vertices in random order.
+
+=back
+
+For the attributes the recommended way is to use the new API.
+
+Do not expect new methods to work for compat02 graphs.
+
+The following compatibility methods exist:
+
+=over 4
+
+=item has_attribute
+
+=item has_attributes
+
+=item get_attribute
+
+=item get_attributes
+
+=item set_attribute
+
+=item set_attributes
+
+=item delete_attribute
+
+=item delete_attributes
+
+Do not use the above, use the new attribute interfaces instead.
+
+=item vertices_unsorted
+
+Alias for vertices() (or rather, vertices05()) since the vertices()
+now always returns the vertices in an unsorted order.  You can also
+use the unsorted_vertices import, but only with a true value (false
+values will cause an error).
+
+=item density_limits
+
+    my ($sparse, $dense, $complete) = $g->density_limits;
+
+Return the "density limits" used to classify graphs as "sparse" or "dense".
+The first limit is C/4 and the second limit is 3C/4, where C is the number
+of edges in a complete graph (the last "limit").
+
+=item density
+
+    my $density = $g->density;
+
+Return the density of the graph, the ratio of the number of edges to the
+number of edges in a complete graph.
+
+=item vertex
+
+    my $v = $g->vertex($v);
+
+Return the vertex if the graph has the vertex, undef otherwise.
+
+=item out_edges
+
+=item in_edges
+
+=item edges($v)
+
+This is now called edges_at($v).
+
+=back
+
+=head2 DIAGNOSTICS
+
+=over 4
+
+=item *
+
+Graph::...Map...: arguments X expected Y ...
+
+If you see these (more user-friendly error messages should have been
+triggered above and before these) please report any such occurrences,
+but in general you should be happy to see these since it means that an
+attempt to call something with a wrong number of arguments was caught
+in time.
+
+=item *
+
+Graph::add_edge: graph is not hyperedged ...
+
+Maybe you used add_weighted_edge() with only the two vertex arguments.
+
+=item *
+
+Not an ARRAY reference at lib/Graph.pm ...
+
+One possibility is that you have code based on Graph 0.2xxxx that
+assumes Graphs being blessed hash references, possibly also assuming
+that certain hash keys are available to use for your own purposes.
+In Graph 0.50 none of this is true.  Please do not expect any
+particular internal implementation of Graphs.  Use inheritance
+and graph/vertex/edge attributes instead.
+
+Another possibility is that you meant to have objects (blessed
+references) as graph vertices, but forgot to use C<refvertexed>
+(see L</refvertexed>) when creating the graph.
+
+=back
+
+=head2 POSSIBLE FUTURES
+
+A possible future direction is a new graph module written for speed:
+this may very possibly mean breaking or limiting some of the APIs or
+behaviour as compared with this release of the module.
+
+What definitely won't happen in future releases is carrying over
+the Graph 0.2xxxx backward compatibility API.
+
+=head1 ACKNOWLEDGEMENTS
+
+All bad terminology, bugs, and inefficiencies are naturally mine, all
+mine, and not the fault of the below.
+
+Thanks to Nathan Goodman and Andras Salamon for bravely betatesting my
+pre-0.50 code.  If they missed something, that was only because of my
+fiendish code.
+
+The following literature for algorithms and some test cases:
+
+=over 4
+
+=item *
+
+Algorithms in C, Third Edition, Part 5, Graph Algorithms, Robert Sedgewick, Addison Wesley
+
+=item *
+
+Introduction to Algorithms, First Edition, Cormen-Leiserson-Rivest, McGraw Hill
+
+=item *
+
+Graphs, Networks and Algorithms, Dieter Jungnickel, Springer
+
+=back
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/AdjacencyMap.pm b/perllib/Graph/AdjacencyMap.pm
new file mode 100644
index 00000000..d2245da6
--- /dev/null
+++ b/perllib/Graph/AdjacencyMap.pm
@@ -0,0 +1,473 @@
+package Graph::AdjacencyMap;
+
+use strict;
+
+require Exporter;
+use vars qw(@ISA @EXPORT_OK %EXPORT_TAGS);
+@ISA = qw(Exporter);
+@EXPORT_OK   = qw(_COUNT _MULTI _COUNTMULTI _GEN_ID
+		  _HYPER _UNORD _UNIQ _REF _UNORDUNIQ _UNIONFIND _LIGHT
+		  _n _f _a _i _s _p _g _u _ni _nc _na _nm);
+%EXPORT_TAGS =
+    (flags =>  [qw(_COUNT _MULTI _COUNTMULTI _GEN_ID
+		   _HYPER _UNORD _UNIQ _REF _UNORDUNIQ _UNIONFIND _LIGHT)],
+     fields => [qw(_n _f _a _i _s _p _g _u _ni _nc _na _nm)]);
+
+sub _COUNT      () {  0x00000001   }
+sub _MULTI      () {  0x00000002   }
+sub _COUNTMULTI () { _COUNT|_MULTI }
+sub _HYPER      () {  0x00000004   }
+sub _UNORD      () {  0x00000008   }
+sub _UNIQ       () {  0x00000010   }
+sub _REF        () {  0x00000020   }
+sub _UNORDUNIQ  () { _UNORD|_UNIQ  }
+sub _UNIONFIND  () {  0x00000040   }
+sub _LIGHT      () {  0x00000080   }
+
+my $_GEN_ID = 0;
+
+sub _GEN_ID () { \$_GEN_ID }
+
+sub _ni () { 0 } # Node index.
+sub _nc () { 1 } # Node count.
+sub _na () { 2 } # Node attributes.
+sub _nm () { 3 } # Node map.
+
+sub _n () { 0 } # Next id.
+sub _f () { 1 } # Flags.
+sub _a () { 2 } # Arity.
+sub _i () { 3 } # Index to path.
+sub _s () { 4 } # Successors / Path to Index.
+sub _p () { 5 } # Predecessors.
+sub _g () { 6 } # Graph (AdjacencyMap::Light)
+
+sub _V () { 2 }  # Graph::_V()
+
+sub _new {
+    my $class = shift;
+    my $map = bless [ 0, @_ ], $class;
+    return $map;
+}
+
+sub _ids {
+    my $m = shift;
+    return $m->[ _i ];
+}
+
+sub has_paths {
+    my $m = shift;
+    return defined $m->[ _i ] && keys %{ $m->[ _i ] };
+}
+
+sub _dump {
+    my $d = Data::Dumper->new([$_[0]],[ref $_[0]]);
+    defined wantarray ? $d->Dump : print $d->Dump;
+}
+
+sub _del_id {
+    my ($m, $i) = @_;
+    my @p = $m->_get_id_path( $i );
+    $m->del_path( @p ) if @p;
+}
+
+sub _new_node {
+    my ($m, $n, $id) = @_;
+    my $f = $m->[ _f ];
+    my $i = $m->[ _n ]++;
+    if (($f & _MULTI)) {
+	$id = 0 if $id eq _GEN_ID;
+	$$n = [ $i, 0, undef, { $id => { } } ];
+    } elsif (($f & _COUNT)) {
+	$$n = [ $i, 1 ];
+    } else {
+	$$n = $i;
+    }
+    return $i;
+}
+
+sub _inc_node {
+    my ($m, $n, $id) = @_;
+    my $f = $m->[ _f ];
+    if (($f & _MULTI)) {
+	if ($id eq _GEN_ID) {
+	    $$n->[ _nc ]++
+		while exists $$n->[ _nm ]->{ $$n->[ _nc ] };
+	    $id = $$n->[ _nc ];
+	}
+	$$n->[ _nm ]->{ $id } = { };
+    } elsif (($f & _COUNT)) {
+	$$n->[ _nc ]++;
+    }
+    return $id;
+}
+
+sub __get_path_node {
+    my $m = shift;
+    my ($p, $k);
+    my $f = $m->[ _f ];
+    @_ = sort @_ if ($f & _UNORD);
+    if ($m->[ _a ] == 2 && @_ == 2 && !($f & (_HYPER|_REF|_UNIQ))) { # Fast path.
+	return unless exists $m->[ _s ]->{ $_[0] };
+	$p = [ $m->[ _s ], $m->[ _s ]->{ $_[0] } ];
+	$k = [ $_[0], $_[1] ];
+    } else {
+	($p, $k) = $m->__has_path( @_ );
+    }
+    return unless defined $p && defined $k;
+    my $l = defined $k->[-1] ? $k->[-1] : "";
+    return ( exists $p->[-1]->{ $l }, $p->[-1]->{ $l }, $p, $k, $l );
+}
+
+sub set_path_by_multi_id {
+    my $m = shift;
+    my ($p, $k) = $m->__set_path( @_ );
+    return unless defined $p && defined $k;
+    my $l = defined $k->[-1] ? $k->[-1] : "";
+    return $m->__set_path_node( $p, $l, @_ );
+}
+
+sub get_multi_ids {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    return () unless ($f & _MULTI);
+    my ($e, $n) = $m->__get_path_node( @_ );
+    return $e ? keys %{ $n->[ _nm ] } : ();
+}
+
+sub _has_path_attrs {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	return keys %{ $p->[-1]->{ $l }->[ _nm ]->{ $id } } ? 1 : 0;
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	return ref $n && $#$n == _na && keys %{ $n->[ _na ] } ? 1 : 0;
+    }
+}
+
+sub _set_path_attrs {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $attr = pop;
+    my $id   = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( @_ );
+    push @_, $id if ($f & _MULTI);
+    my ($p, $k) = $m->__set_path( @_ );
+    return unless defined $p && defined $k;
+    my $l = defined $k->[-1] ? $k->[-1] : "";
+    $m->__set_path_node( $p, $l, @_ ) unless exists $p->[-1]->{ $l };
+    if (($f & _MULTI)) {
+	$p->[-1]->{ $l }->[ _nm ]->{ $id } = $attr;
+    } else {
+	# Extend the node if it is a simple id node.
+	$p->[-1]->{ $l } = [ $p->[-1]->{ $l }, 1 ] unless ref $p->[-1]->{ $l };
+	$p->[-1]->{ $l }->[ _na ] = $attr;
+    }
+}
+
+sub _has_path_attr {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $attr = pop;
+    my $id   = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	exists $p->[-1]->{ $l }->[ _nm ]->{ $id }->{ $attr };
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	return ref $n && $#$n == _na ? exists $n->[ _na ]->{ $attr } : undef;
+    }
+}
+
+sub _set_path_attr {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $val  = pop;
+    my $attr = pop;
+    my $id   = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    my ($p, $k);
+    $m->__attr( \@_ ); # _LIGHT maps need this to get upgraded when needed.
+    push @_, $id if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    if ($m->[ _a ] == 2 && @_ == 2 && !($f & (_REF|_UNIQ|_HYPER|_UNIQ))) {
+	$m->[ _s ]->{ $_[0] } ||= { };
+	$p = [ $m->[ _s ], $m->[ _s ]->{ $_[0] } ];
+	$k = [ $_[0], $_[1] ];
+    } else {
+	($p, $k) = $m->__set_path( @_ );
+    }
+    return unless defined $p && defined $k;
+    my $l = defined $k->[-1] ? $k->[-1] : "";
+    $m->__set_path_node( $p, $l, @_ ) unless exists $p->[-1]->{ $l };
+    if (($f & _MULTI)) {
+	$p->[-1]->{ $l }->[ _nm ]->{ $id }->{ $attr } = $val;
+    } else {
+	# Extend the node if it is a simple id node.
+	$p->[-1]->{ $l } = [ $p->[-1]->{ $l }, 1 ] unless ref $p->[-1]->{ $l };
+	$p->[-1]->{ $l }->[ _na ]->{ $attr } = $val;
+    }
+    return $val;
+}
+
+sub _get_path_attrs {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id   = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	$p->[-1]->{ $l }->[ _nm ]->{ $id };
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return unless $e;
+	return $n->[ _na ] if ref $n && $#$n == _na;
+	return;
+    }
+}
+
+sub _get_path_attr {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $attr = pop;
+    my $id = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	return $p->[-1]->{ $l }->[ _nm ]->{ $id }->{ $attr };
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	return ref $n && $#$n == _na ? $n->[ _na ]->{ $attr } : undef;
+    }
+}
+
+sub _get_path_attr_names {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	keys %{ $p->[-1]->{ $l }->[ _nm ]->{ $id } };
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	return keys %{ $n->[ _na ] } if ref $n && $#$n == _na;
+	return;
+    }
+}
+
+sub _get_path_attr_values {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	values %{ $p->[-1]->{ $l }->[ _nm ]->{ $id } };
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	return values %{ $n->[ _na ] } if ref $n && $#$n == _na;
+	return;
+    }
+}
+
+sub _del_path_attrs {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	delete $p->[-1]->{ $l }->[ _nm ]->{ $id };
+	unless (keys %{ $p->[-1]->{ $l }->[ _nm ] } ||
+		(defined $p->[-1]->{ $l }->[ _na ] &&
+		 keys %{ $p->[-1]->{ $l }->[ _na ] })) {
+	    delete $p->[-1]->{ $l };
+	}
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	if (ref $n) {
+	    $e = _na == $#$n && keys %{ $n->[ _na ] } ? 1 : 0;
+	    $#$n = _na - 1;
+	    return $e;
+	} else {
+	    return 0;
+	}
+    }
+}
+
+sub _del_path_attr {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $attr = pop;
+    my $id = pop if ($f & _MULTI);
+    @_ = sort @_ if ($f & _UNORD);
+    $m->__attr( \@_ );
+    if (($f & _MULTI)) {
+	my ($p, $k) = $m->__has_path( @_ );
+	return unless defined $p && defined $k;
+	my $l = defined $k->[-1] ? $k->[-1] : "";
+	delete $p->[-1]->{ $l }->[ _nm ]->{ $id }->{ $attr };
+	$m->_del_path_attrs( @_, $id )
+	    unless keys %{ $p->[-1]->{ $l }->[ _nm ]->{ $id } };
+    } else {
+	my ($e, $n) = $m->__get_path_node( @_ );
+	return undef unless $e;
+	if (ref $n && $#$n == _na && exists $n->[ _na ]->{ $attr }) {
+	    delete $n->[ _na ]->{ $attr };
+	    return 1;
+	} else {
+	    return 0;
+	}
+    }
+}
+
+sub _is_COUNT { $_[0]->[ _f ] & _COUNT }
+sub _is_MULTI { $_[0]->[ _f ] & _MULTI }
+sub _is_HYPER { $_[0]->[ _f ] & _HYPER }
+sub _is_UNORD { $_[0]->[ _f ] & _UNORD }
+sub _is_UNIQ  { $_[0]->[ _f ] & _UNIQ  }
+sub _is_REF   { $_[0]->[ _f ] & _REF   }
+
+sub __arg {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my @a = @{$_[0]};
+    if ($f & _UNIQ) {
+	my %u;
+	if ($f & _UNORD) {
+	    @u{ @a } = @a;
+	    @a = values %u;
+	} else {
+	    my @u;
+	    for my $e (@a) {
+		push @u, $e if $u{$e}++ == 0;
+	    }
+	    @a = @u;
+	}
+    }
+    # Alphabetic or numeric sort, does not matter as long as it unifies.
+    @{$_[0]} = ($f & _UNORD) ? sort @a : @a;
+}
+
+sub _successors {
+    my $E = shift;
+    my $g = shift;
+    my $V = $g->[ _V ];
+    map { my @v = @{ $_->[ 1 ] };
+	  shift @v;
+	  map { $V->_get_id_path($_) } @v } $g->_edges_from( @_ );
+}
+
+sub _predecessors {
+    my $E = shift;
+    my $g = shift;
+    my $V = $g->[ _V ];
+    if (wantarray) {
+	map { my @v = @{ $_->[ 1 ] };
+	      pop @v;
+	      map { $V->_get_id_path($_) } @v } $g->_edges_to( @_ );
+    } else {
+	return $g->_edges_to( @_ );
+    }
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::AdjacencyMap - create and a map of graph vertices or edges
+
+=head1 SYNOPSIS
+
+    Internal.
+
+=head1 DESCRIPTION
+
+B<This module is meant for internal use by the Graph module.>
+
+=head2 Object Methods
+
+=over 4
+
+=item del_path(@id)
+
+Delete a Map path by ids.
+
+=item del_path_by_multi_id($id)
+
+Delete a Map path by a multi(vertex) id.
+
+=item get_multi_ids
+
+Return the multi ids.
+
+=item has_path(@id)
+
+Return true if the Map has the path by ids, false if not.
+
+=item has_paths
+
+Return true if the Map has any paths, false if not.
+
+=item has_path_by_multi_id($id)
+
+Return true ifd the a Map has the path by a multi(vertex) id, false if not.
+
+=item paths
+
+Return all the paths of the Map.
+
+=item set_path(@id)
+
+Set the path by @ids.
+
+=item set_path_by_multi_id
+
+Set the path in the Map by the multi id.
+
+=back
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/AdjacencyMap/Heavy.pm b/perllib/Graph/AdjacencyMap/Heavy.pm
new file mode 100644
index 00000000..262bd4f5
--- /dev/null
+++ b/perllib/Graph/AdjacencyMap/Heavy.pm
@@ -0,0 +1,253 @@
+package Graph::AdjacencyMap::Heavy;
+
+# THIS IS INTERNAL IMPLEMENTATION ONLY, NOT TO BE USED DIRECTLY.
+# THE INTERFACE IS HARD TO USE AND GOING TO STAY THAT WAY AND
+# ALMOST GUARANTEED TO CHANGE OR GO AWAY IN FUTURE RELEASES.
+
+use strict;
+
+# $SIG{__DIE__ } = sub { use Carp; confess };
+# $SIG{__WARN__} = sub { use Carp; confess };
+
+use Graph::AdjacencyMap qw(:flags :fields);
+use base 'Graph::AdjacencyMap';
+
+require overload; # for de-overloading
+
+require Data::Dumper;
+
+sub __set_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    if (@_ != $m->[ _a ] && !($f & _HYPER)) {
+	require Carp;
+	Carp::confess(sprintf "Graph::AdjacencyMap::Heavy: arguments %d expected %d",
+		      scalar @_, $m->[ _a ]);
+    }
+    my $p;
+    $p = ($f & _HYPER) ?
+	(( $m->[ _s ] ||= [ ] )->[ @_ ] ||= { }) :
+	(  $m->[ _s ]                   ||= { });
+    my @p = $p;
+    my @k;
+    while (@_) {
+	my $k = shift;
+	my $q = ref $k && ($f & _REF) && overload::Method($k, '""') ? overload::StrVal($k) : $k;
+	if (@_) {
+	    $p = $p->{ $q } ||= {};
+	    return unless $p;
+	    push @p, $p;
+	}
+	push @k, $q;
+    }
+    return (\@p, \@k);
+}
+
+sub __set_path_node {
+    my ($m, $p, $l) = splice @_, 0, 3;
+    my $f = $m->[ _f ] ;
+    my $id = pop if ($f & _MULTI);
+    unless (exists $p->[-1]->{ $l }) {
+	my $i = $m->_new_node( \$p->[-1]->{ $l }, $id );
+	$m->[ _i ]->{ defined $i ? $i : "" } = [ @_ ];
+        return defined $id ? ($id eq _GEN_ID ? $$id : $id) : $i;
+    } else {
+	return $m->_inc_node( \$p->[-1]->{ $l }, $id );
+    }
+}
+
+sub set_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+        else { $m->__arg(\@_) }
+    }
+    my ($p, $k) = $m->__set_path( @_ );
+    return unless defined $p && defined $k;
+    my $l = defined $k->[-1] ? $k->[-1] : "";
+    return $m->__set_path_node( $p, $l, @_ );
+}
+
+sub __has_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    if (@_ != $m->[ _a ] && !($f & _HYPER)) {
+	require Carp;
+	Carp::confess(sprintf "Graph::AdjacencyMap::Heavy: arguments %d expected %d",
+		      scalar @_, $m->[ _a ]);
+    }
+    if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+        else { $m->__arg(\@_) }
+    }
+    my $p = $m->[ _s ];
+    return unless defined $p;
+    $p = $p->[ @_ ] if ($f & _HYPER);
+    return unless defined $p;
+    my @p = $p;
+    my @k;
+    while (@_) {
+	my $k = shift;
+	my $q = ref $k && ($f & _REF) && overload::Method($k, '""') ? overload::StrVal($k) : $k;
+	if (@_) {
+	    $p = $p->{ $q };
+	    return unless defined $p;
+	    push @p, $p;
+	}
+	push @k, $q;
+    }
+    return (\@p, \@k);
+}
+
+sub has_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+        else { $m->__arg(\@_) }
+    }
+    my ($p, $k) = $m->__has_path( @_ );
+    return unless defined $p && defined $k;
+    return exists $p->[-1]->{ defined $k->[-1] ? $k->[-1] : "" };
+}
+
+sub has_path_by_multi_id {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop;
+    if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+        else { $m->__arg(\@_) }
+    }
+    my ($e, $n) = $m->__get_path_node( @_ );
+    return undef unless $e;
+    return exists $n->[ _nm ]->{ $id };
+}
+
+sub _get_path_node {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    if ($m->[ _a ] == 2 && @_ == 2 && !($f & (_HYPER|_REF|_UNIQ))) { # Fast path.
+	@_ = sort @_ if ($f & _UNORD);
+	return unless exists $m->[ _s ]->{ $_[0] };
+	my $p = [ $m->[ _s ], $m->[ _s ]->{ $_[0] } ];
+	my $k = [ $_[0], $_[1] ];
+	my $l = $_[1];
+	return ( exists $p->[-1]->{ $l }, $p->[-1]->{ $l }, $p, $k, $l );
+    } else {
+	if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	    if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+	    else { $m->__arg(\@_) }
+	}
+	$m->__get_path_node( @_ );
+    }
+}
+
+sub _get_path_id {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my ($e, $n);
+    if ($m->[ _a ] == 2 && @_ == 2 && !($f & (_HYPER|_REF|_UNIQ))) { # Fast path.
+	@_ = sort @_ if ($f & _UNORD);
+	return unless exists $m->[ _s ]->{ $_[0] };
+	my $p = $m->[ _s ]->{ $_[0] };
+	$e = exists $p->{ $_[1] };
+	$n = $p->{ $_[1] };
+    } else {
+	($e, $n) = $m->_get_path_node( @_ );
+    }
+    return undef unless $e;
+    return ref $n ? $n->[ _ni ] : $n;
+}
+
+sub _get_path_count {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my ($e, $n) = $m->_get_path_node( @_ );
+    return undef unless $e && defined $n;
+    return
+	($f & _COUNT) ? $n->[ _nc ] :
+	($f & _MULTI) ? scalar keys %{ $n->[ _nm ] } : 1;
+}
+
+sub __attr {
+    my $m = shift;
+    if (@_) {
+	if (ref $_[0] && @{ $_[0] }) {
+	    if (@{ $_[0] } != $m->[ _a ]) {
+		require Carp;
+		Carp::confess(sprintf
+			      "Graph::AdjacencyMap::Heavy: arguments %d expected %d\n",
+			      scalar @{ $_[0] }, $m->[ _a ]);
+	    }
+	    my $f = $m->[ _f ];
+	    if (@{ $_[0] } > 1 && ($f & _UNORDUNIQ)) {
+		if (($f & _UNORDUNIQ) == _UNORD && @{ $_[0] } == 2) {
+		    @{ $_[0] } = sort @{ $_[0] }
+		} else { $m->__arg(\@_) }
+	    }
+	}
+    }
+}
+
+sub _get_id_path {
+    my ($m, $i) = @_;
+    my $p = defined $i ? $m->[ _i ]->{ $i } : undef;
+    return defined $p ? @$p : ( );
+}
+
+sub del_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+        else { $m->__arg(\@_) }
+    }
+    my ($e, $n, $p, $k, $l) = $m->__get_path_node( @_ );
+    return unless $e;
+    my $c = ($f & _COUNT) ? --$n->[ _nc ] : 0;
+    if ($c == 0) {
+	delete $m->[ _i ]->{ ref $n ? $n->[ _ni ] : $n };
+	delete $p->[-1]->{ $l };
+	while (@$p && @$k && keys %{ $p->[-1]->{ $k->[-1] } } == 0) {
+	    delete $p->[-1]->{ $k->[-1] };
+	    pop @$p;
+	    pop @$k;
+	}
+    }
+    return 1;
+}
+
+sub del_path_by_multi_id {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop;
+    if (@_ > 1 && ($f & _UNORDUNIQ)) {
+	if (($f & _UNORDUNIQ) == _UNORD && @_ == 2) { @_ = sort @_ }
+        else { $m->__arg(\@_) }
+    }
+    my ($e, $n, $p, $k, $l) = $m->__get_path_node( @_ );
+    return unless $e;
+    delete $n->[ _nm ]->{ $id };
+    unless (keys %{ $n->[ _nm ] }) {
+	delete $m->[ _i ]->{ $n->[ _ni ] };
+	delete $p->[-1]->{ $l };
+	while (@$p && @$k && keys %{ $p->[-1]->{ $k->[-1] } } == 0) {
+	    delete $p->[-1]->{ $k->[-1] };
+	    pop @$p;
+	    pop @$k;
+	}
+    }
+    return 1;
+}
+
+sub paths {
+    my $m = shift;
+    return values %{ $m->[ _i ] } if defined $m->[ _i ];
+    wantarray ? ( ) : 0;
+}
+
+1;
+__END__
diff --git a/perllib/Graph/AdjacencyMap/Light.pm b/perllib/Graph/AdjacencyMap/Light.pm
new file mode 100644
index 00000000..bedaf652
--- /dev/null
+++ b/perllib/Graph/AdjacencyMap/Light.pm
@@ -0,0 +1,247 @@
+package Graph::AdjacencyMap::Light;
+
+# THIS IS INTERNAL IMPLEMENTATION ONLY, NOT TO BE USED DIRECTLY.
+# THE INTERFACE IS HARD TO USE AND GOING TO STAY THAT WAY AND
+# ALMOST GUARANTEED TO CHANGE OR GO AWAY IN FUTURE RELEASES.
+
+use strict;
+
+use Graph::AdjacencyMap qw(:flags :fields);
+use base 'Graph::AdjacencyMap';
+
+use Scalar::Util qw(weaken);
+
+use Graph::AdjacencyMap::Heavy;
+use Graph::AdjacencyMap::Vertex;
+
+sub _V () { 2 } # Graph::_V
+sub _E () { 3 } # Graph::_E
+sub _F () { 0 } # Graph::_F
+
+sub _new {
+    my ($class, $graph, $flags, $arity) = @_;
+    my $m = bless [ ], $class;
+    $m->[ _n ] = 0;
+    $m->[ _f ] = $flags | _LIGHT;
+    $m->[ _a ] = $arity;
+    $m->[ _i ] = { };
+    $m->[ _s ] = { };
+    $m->[ _p ] = { };
+    $m->[ _g ] = $graph;
+    weaken $m->[ _g ]; # So that DESTROY finds us earlier.
+    return $m;
+}
+
+sub set_path {
+    my $m = shift;
+    my ($n, $f, $a, $i, $s, $p) = @$m;
+    if ($a == 2) {
+	@_ = sort @_ if ($f & _UNORD);
+    }
+    my $e0 = shift;
+    if ($a == 2) {
+	my $e1 = shift;
+	unless (exists $s->{ $e0 } && exists $s->{ $e0 }->{ $e1 }) {
+	    $n = $m->[ _n ]++;
+	    $i->{ $n } = [ $e0, $e1 ];
+	    $s->{ $e0 }->{ $e1 } = $n;
+	    $p->{ $e1 }->{ $e0 } = $n;
+	}
+    } else {
+	unless (exists $s->{ $e0 }) {
+	    $n = $m->[ _n ]++;
+	    $s->{ $e0 } = $n;
+	    $i->{ $n } = $e0;
+	}
+    }
+}
+
+sub has_path {
+    my $m = shift;
+    my ($n, $f, $a, $i, $s) = @$m;
+    return 0 unless $a == @_;
+    my $e;
+    if ($a == 2) {
+	@_ = sort @_ if ($f & _UNORD);
+	$e = shift;
+	return 0 unless exists $s->{ $e };
+        $s = $s->{ $e };
+    }
+    $e = shift;
+    exists $s->{ $e };
+}
+
+sub _get_path_id {
+    my $m = shift;
+    my ($n, $f, $a, $i, $s) = @$m;
+    return undef unless $a == @_;
+    my $e;
+    if ($a == 2) {
+	@_ = sort @_ if ($f & _UNORD);
+	$e = shift;
+	return undef unless exists $s->{ $e };
+        $s = $s->{ $e };
+    }
+    $e = shift;
+    $s->{ $e };
+}
+
+sub _get_path_count {
+    my $m = shift;
+    my ($n, $f, $a, $i, $s) = @$m;
+    my $e;
+    if (@_ == 2) {
+	@_ = sort @_ if ($f & _UNORD);
+	$e = shift;
+	return undef unless exists $s->{ $e };
+        $s = $s->{ $e };
+    }
+    $e = shift;
+    return exists $s->{ $e } ? 1 : 0;
+}
+
+sub has_paths {
+    my $m = shift;
+    my ($n, $f, $a, $i, $s) = @$m;
+    keys %$s;
+}
+
+sub paths {
+    my $m = shift;
+    my ($n, $f, $a, $i) = @$m;
+    if (defined $i) {
+	my ($k, $v) = each %$i;
+	if (ref $v) {
+	    return values %{ $i };
+	} else {
+	    return map { [ $_ ] } values %{ $i };
+	}
+    } else {
+	return ( );
+    }
+}
+
+sub _get_id_path {
+    my $m = shift;
+    my ($n, $f, $a, $i) = @$m;
+    my $p = $i->{ $_[ 0 ] };
+    defined $p ? ( ref $p eq 'ARRAY' ? @$p : $p ) : ( );
+}
+
+sub del_path {
+    my $m = shift;
+    my ($n, $f, $a, $i, $s, $p) = @$m;
+    if (@_ == 2) {
+	@_ = sort @_ if ($f & _UNORD);
+	my $e0 = shift;
+	return 0 unless exists $s->{ $e0 };
+	my $e1 = shift;
+	if (defined($n = $s->{ $e0 }->{ $e1 })) {
+	    delete $i->{ $n };
+            delete $s->{ $e0 }->{ $e1 };
+            delete $p->{ $e1 }->{ $e0 };
+	    delete $s->{ $e0 } unless keys %{ $s->{ $e0 } };
+	    delete $p->{ $e1 } unless keys %{ $p->{ $e1 } };
+	    return 1;
+	}
+    } else {
+	my $e = shift;
+	if (defined($n = $s->{ $e })) {
+	    delete $i->{ $n };
+	    delete $s->{ $e };
+	    return 1;
+	}
+    }
+    return 0;
+}
+
+sub __successors {
+    my $E = shift;
+    return wantarray ? () : 0 unless defined $E->[ _s ];
+    my $g = shift;
+    my $V = $g->[ _V ];
+    return wantarray ? () : 0 unless defined $V && defined $V->[ _s ];
+    # my $i = $V->_get_path_id( $_[0] );
+    my $i =
+	($V->[ _f ] & _LIGHT) ?
+	    $V->[ _s ]->{ $_[0] } :
+	    $V->_get_path_id( $_[0] );
+    return wantarray ? () : 0 unless defined $i && defined $E->[ _s ]->{ $i };
+    return keys %{ $E->[ _s ]->{ $i } };
+}
+
+sub _successors {
+    my $E = shift;
+    my $g = shift;
+    my @s = $E->__successors($g, @_);
+    if (($E->[ _f ] & _UNORD)) {
+	push @s, $E->__predecessors($g, @_);
+	my %s; @s{ @s } = ();
+	@s = keys %s;
+    }
+    my $V = $g->[ _V ];
+    return wantarray ? map { $V->[ _i ]->{ $_ } } @s : @s;
+}
+
+sub __predecessors {
+    my $E = shift;
+    return wantarray ? () : 0 unless defined $E->[ _p ];
+    my $g = shift;
+    my $V = $g->[ _V ];
+    return wantarray ? () : 0 unless defined $V && defined $V->[ _s ];
+    # my $i = $V->_get_path_id( $_[0] );
+    my $i =
+	($V->[ _f ] & _LIGHT) ?
+	    $V->[ _s ]->{ $_[0] } :
+	    $V->_get_path_id( $_[0] );
+    return wantarray ? () : 0 unless defined $i && defined $E->[ _p ]->{ $i };
+    return keys %{ $E->[ _p ]->{ $i } };
+}
+
+sub _predecessors {
+    my $E = shift;
+    my $g = shift;
+    my @p = $E->__predecessors($g, @_);
+    if ($E->[ _f ] & _UNORD) {
+	push @p, $E->__successors($g, @_);
+	my %p; @p{ @p } = ();
+	@p = keys %p;
+    }
+    my $V = $g->[ _V ];
+    return wantarray ? map { $V->[ _i ]->{ $_ } } @p : @p;
+}
+
+sub __attr {
+    # Major magic takes place here: we rebless the appropriate 'light'
+    # map into a more complex map and then redispatch the method.
+    my $m = $_[0];
+    my ($n, $f, $a, $i, $s, $p, $g) = @$m;
+    my ($k, $v) = each %$i;
+    my @V = @{ $g->[ _V ] };
+    my @E = $g->edges; # TODO: Both these (ZZZ) lines are mysteriously needed!
+    # ZZZ: an example of failing tests is t/52_edge_attributes.t.
+    if (ref $v eq 'ARRAY') { # Edges, then.
+	# print "Reedging.\n";
+	@E = $g->edges; # TODO: Both these (ZZZ) lines are mysteriously needed!
+	$g->[ _E ] = $m = Graph::AdjacencyMap::Heavy->_new($f, 2);
+	$g->add_edges( @E );
+    } else {
+	# print "Revertexing.\n";
+	$m = Graph::AdjacencyMap::Vertex->_new(($f & ~_LIGHT), 1);
+	$m->[ _n ] = $V[ _n ];
+	$m->[ _i ] = $V[ _i ];
+	$m->[ _s ] = $V[ _s ];
+	$m->[ _p ] = $V[ _p ];
+	$g->[ _V ] = $m;
+    }
+    $_[0] = $m;
+    goto &{ ref($m) . "::__attr" }; # Redispatch.
+}
+
+sub _is_COUNT    () { 0 }
+sub _is_MULTI    () { 0 }
+sub _is_HYPER    () { 0 }
+sub _is_UNIQ     () { 0 }
+sub _is_REF      () { 0 }
+
+1;
diff --git a/perllib/Graph/AdjacencyMap/Vertex.pm b/perllib/Graph/AdjacencyMap/Vertex.pm
new file mode 100644
index 00000000..72d81427
--- /dev/null
+++ b/perllib/Graph/AdjacencyMap/Vertex.pm
@@ -0,0 +1,216 @@
+package Graph::AdjacencyMap::Vertex;
+
+# THIS IS INTERNAL IMPLEMENTATION ONLY, NOT TO BE USED DIRECTLY.
+# THE INTERFACE IS HARD TO USE AND GOING TO STAY THAT WAY AND
+# ALMOST GUARANTEED TO CHANGE OR GO AWAY IN FUTURE RELEASES.
+
+use strict;
+
+# $SIG{__DIE__ } = sub { use Carp; confess };
+# $SIG{__WARN__} = sub { use Carp; confess };
+
+use Graph::AdjacencyMap qw(:flags :fields);
+use base 'Graph::AdjacencyMap';
+
+use Scalar::Util qw(weaken);
+
+sub _new {
+    my ($class, $flags, $arity) = @_;
+    bless [ 0, $flags, $arity ], $class;
+}
+
+require overload; # for de-overloading
+
+sub __set_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    if (@_ != 1) {
+	require Carp;
+	Carp::confess(sprintf "Graph::AdjacencyMap::Vertex: arguments %d expected 1", scalar @_);
+    }
+    my $p;
+    $p = $m->[ _s ] ||= { };
+    my @p = $p;
+    my @k;
+    my $k = shift;
+    my $q = ref $k && ($f & _REF) && overload::Method($k, '""') ? overload::StrVal($k) : $k;
+    push @k, $q;
+    return (\@p, \@k);
+}
+
+sub __set_path_node {
+    my ($m, $p, $l) = splice @_, 0, 3;
+    my $f = $m->[ _f ];
+    my $id = pop if ($f & _MULTI);
+    unless (exists $p->[-1]->{ $l }) {
+	my $i = $m->_new_node( \$p->[-1]->{ $l }, $id );
+	$m->[ _i ]->{ defined $i ? $i : "" } = $_[0];
+    } else {
+	$m->_inc_node( \$p->[-1]->{ $l }, $id );
+    }
+}
+
+sub set_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my ($p, $k) = $m->__set_path( @_ );
+    return unless defined $p && defined $k;
+    my $l = defined $k->[-1] ? $k->[-1] : "";
+    my $set = $m->__set_path_node( $p, $l, @_ );
+    return $set;
+}
+
+sub __has_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    if (@_ != 1) {
+	require Carp;
+	Carp::confess(sprintf
+		      "Graph::AdjacencyMap: arguments %d expected 1\n",
+		      scalar @_);
+    }
+    my $p = $m->[ _s ];
+    return unless defined $p;
+    my @p = $p;
+    my @k;
+    my $k = shift;
+    my $q = ref $k && ($f & _REF) && overload::Method($k, '""') ? overload::StrVal($k) : $k;
+    push @k, $q;
+    return (\@p, \@k);
+}
+
+sub has_path {
+    my $m = shift;
+    my ($p, $k) = $m->__has_path( @_ );
+    return unless defined $p && defined $k;
+    return exists $p->[-1]->{ defined $k->[-1] ? $k->[-1] : "" };
+}
+
+sub has_path_by_multi_id {
+    my $m = shift;
+    my $id = pop;
+    my ($e, $n) = $m->__get_path_node( @_ );
+    return undef unless $e;
+    return exists $n->[ _nm ]->{ $id };
+}
+
+sub _get_path_id {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my ($e, $n) = $m->__get_path_node( @_ );
+    return undef unless $e;
+    return ref $n ? $n->[ _ni ] : $n;
+}
+
+sub _get_path_count {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my ($e, $n) = $m->__get_path_node( @_ );
+    return 0 unless $e && defined $n;
+    return
+	($f & _COUNT) ? $n->[ _nc ] :
+	($f & _MULTI) ? scalar keys %{ $n->[ _nm ] } : 1;
+}
+
+sub __attr {
+    my $m = shift;
+    if (@_ && ref $_[0] && @{ $_[0] } != $m->[ _a ]) {
+	require Carp;
+	Carp::confess(sprintf "Graph::AdjacencyMap::Vertex: arguments %d expected %d",
+		      scalar @{ $_[0] }, $m->[ _a ]);
+    }
+}
+
+sub _get_id_path {
+    my ($m, $i) = @_;
+    return defined $m->[ _i ] ? $m->[ _i ]->{ $i } : undef;
+}
+
+sub del_path {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my ($e, $n, $p, $k, $l) = $m->__get_path_node( @_ );
+    return unless $e;
+    my $c = ($f & _COUNT) ? --$n->[ _nc ] : 0;
+    if ($c == 0) {
+	delete $m->[ _i ]->{ ref $n ? $n->[ _ni ] : $n };
+	delete $p->[ -1 ]->{ $l };
+    }
+    return 1;
+}
+
+sub del_path_by_multi_id {
+    my $m = shift;
+    my $f = $m->[ _f ];
+    my $id = pop;
+    my ($e, $n, $p, $k, $l) = $m->__get_path_node( @_ );
+    return unless $e;
+    delete $n->[ _nm ]->{ $id };
+    unless (keys %{ $n->[ _nm ] }) {
+	delete $m->[ _i ]->{ $n->[ _ni ] };
+	delete $p->[-1]->{ $l };
+    }
+    return 1;
+}
+
+sub paths {
+    my $m = shift;
+    return map { [ $_ ] } values %{ $m->[ _i ] } if defined $m->[ _i ];
+    wantarray ? ( ) : 0;
+}
+
+1;
+=pod
+
+=head1 NAME
+
+Graph::AdjacencyMap - create and a map of graph vertices or edges
+
+=head1 SYNOPSIS
+
+    Internal.
+
+=head1 DESCRIPTION
+
+B<This module is meant for internal use by the Graph module.>
+
+=head2 Object Methods
+
+=over 4
+
+=item del_path(@id)
+
+Delete a Map path by ids.
+
+=item del_path_by_multi_id($id)
+
+Delete a Map path by a multi(vertex) id.
+
+=item has_path(@id)
+
+Return true if the Map has the path by ids, false if not.
+
+=item has_path_by_multi_id($id)
+
+Return true ifd the a Map has the path by a multi(vertex) id, false if not.
+
+=item paths
+
+Return all the paths of the Map.
+
+=item set_path(@id)
+
+Set the path by @ids.
+
+=back
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/AdjacencyMatrix.pm b/perllib/Graph/AdjacencyMatrix.pm
new file mode 100644
index 00000000..6c648fec
--- /dev/null
+++ b/perllib/Graph/AdjacencyMatrix.pm
@@ -0,0 +1,223 @@
+package Graph::AdjacencyMatrix;
+
+use strict;
+
+use Graph::BitMatrix;
+use Graph::Matrix;
+
+use base 'Graph::BitMatrix';
+
+use Graph::AdjacencyMap qw(:flags :fields);
+
+sub _V () { 2 } # Graph::_V
+sub _E () { 3 } # Graph::_E
+
+sub new {
+    my ($class, $g, %opt) = @_;
+    my $n;
+    my @V = $g->vertices;
+    my $want_distance;
+    if (exists $opt{distance_matrix}) {
+	$want_distance = $opt{distance_matrix};
+	delete $opt{distance_matrix};
+    }
+    my $d = Graph::_defattr();
+    if (exists $opt{attribute_name}) {
+	$d = $opt{attribute_name};
+	$want_distance++;
+    }
+    delete $opt{attribute_name};
+    my $want_transitive = 0;
+    if (exists $opt{is_transitive}) {
+	$want_transitive = $opt{is_transitive};
+	delete $opt{is_transitive};
+    }
+    Graph::_opt_unknown(\%opt);
+    if ($want_distance) {
+	$n = Graph::Matrix->new($g);
+	for my $v (@V) { $n->set($v, $v, 0) }
+    }
+    my $m = Graph::BitMatrix->new($g, connect_edges => $want_distance);
+    if ($want_distance) {
+	# for my $u (@V) {
+	#     for my $v (@V) {
+	#         if ($g->has_edge($u, $v)) {
+	#             $n->set($u, $v,
+	#                    $g->get_edge_attribute($u, $v, $d));
+        #        }
+        #     }
+        # }
+	my $Vi = $g->[_V]->[_i];
+	my $Ei = $g->[_E]->[_i];
+	my %V; @V{ @V } = 0 .. $#V;
+	my $n0 = $n->[0];
+	my $n1 = $n->[1];
+	if ($g->is_undirected) {
+	    for my $e (keys %{ $Ei }) {
+		my ($i0, $j0) = @{ $Ei->{ $e } };
+		my $i1 = $V{ $Vi->{ $i0 } };
+		my $j1 = $V{ $Vi->{ $j0 } };
+		my $u = $V[ $i1 ];
+		my $v = $V[ $j1 ];
+		$n0->[ $i1 ]->[ $j1 ] = 
+		    $g->get_edge_attribute($u, $v, $d);
+		$n0->[ $j1 ]->[ $i1 ] =
+		    $g->get_edge_attribute($v, $u, $d);
+	    }
+	} else {
+	    for my $e (keys %{ $Ei }) {
+		my ($i0, $j0) = @{ $Ei->{ $e } };
+		my $i1 = $V{ $Vi->{ $i0 } };
+		my $j1 = $V{ $Vi->{ $j0 } };
+		my $u = $V[ $i1 ];
+		my $v = $V[ $j1 ];
+		$n0->[ $i1 ]->[ $j1 ] =
+		    $g->get_edge_attribute($u, $v, $d);
+	    }
+	}
+    }
+    bless [ $m, $n, [ @V ] ], $class;
+}
+
+sub adjacency_matrix {
+    my $am = shift;
+    $am->[0];
+}
+
+sub distance_matrix {
+    my $am = shift;
+    $am->[1];
+}
+
+sub vertices {
+    my $am = shift;
+    @{ $am->[2] };
+}
+
+sub is_adjacent {
+    my ($m, $u, $v) = @_;
+    $m->[0]->get($u, $v) ? 1 : 0;
+}
+
+sub distance {
+    my ($m, $u, $v) = @_;
+    defined $m->[1] ? $m->[1]->get($u, $v) : undef;
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::AdjacencyMatrix - create and query the adjacency matrix of graph G
+
+=head1 SYNOPSIS
+
+    use Graph::AdjacencyMatrix;
+    use Graph::Directed; # or Undirected
+
+    my $g  = Graph::Directed->new;
+    $g->add_...(); # build $g
+
+    my $am = Graph::AdjacencyMatrix->new($g);
+    $am->is_adjacent($u, $v)
+
+    my $am = Graph::AdjacencyMatrix->new($g, distance_matrix => 1);
+    $am->distance($u, $v)
+
+    my $am = Graph::AdjacencyMatrix->new($g, attribute_name => 'length');
+    $am->distance($u, $v)
+
+    my $am = Graph::AdjacencyMatrix->new($g, ...);
+    my @V  = $am->vertices();
+
+=head1 DESCRIPTION
+
+You can use C<Graph::AdjacencyMatrix> to compute the adjacency matrix
+and optionally also the distance matrix of a graph, and after that
+query the adjacencyness between vertices by using the C<is_adjacent()>
+method, or query the distance between vertices by using the
+C<distance()> method.
+
+By default the edge attribute used for distance is C<w>, but you
+can change that in new(), see below.
+
+If you modify the graph after creating the adjacency matrix of it,
+the adjacency matrix and the distance matrix may become invalid.
+
+=head1 Methods
+
+=head2 Class Methods
+
+=over 4
+
+=item new($g)
+
+Construct the adjacency matrix of the graph $g.
+
+=item new($g, options)
+
+Construct the adjacency matrix of the graph $g with options as a hash.
+The known options are
+
+=over 8
+
+=item distance_matrix => boolean
+
+By default only the adjacency matrix is computed.  To compute also the
+distance matrix, use the attribute C<distance_matrix> with a true value
+to the new() constructor.
+
+=item attribute_name => attribute_name
+
+By default the edge attribute used for distance is C<w>.  You can
+change that by giving another attribute name with the C<attribute_name>
+attribute to new() constructor.  Using this attribute also implicitly
+causes the distance matrix to be computed.
+
+=back
+
+=back
+
+=head2 Object Methods
+
+=over 4
+
+=item is_adjacent($u, $v)
+
+Return true if the vertex $v is adjacent to vertex $u, or false if not.
+
+=item distance($u, $v)
+
+Return the distance between the vertices $u and $v, or C<undef> if
+the vertices are not adjacent.
+
+=item adjacency_matrix
+
+Return the adjacency matrix itself (a list of bitvector scalars).
+
+=item vertices
+
+Return the list of vertices (useful for indexing the adjacency matrix).
+
+=back
+
+=head1 ALGORITHM
+
+The algorithm used to create the matrix is two nested loops, which is
+O(V**2) in time, and the returned matrices are O(V**2) in space.
+
+=head1 SEE ALSO
+
+L<Graph::TransitiveClosure>, L<Graph::BitMatrix>
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/Attribute.pm b/perllib/Graph/Attribute.pm
new file mode 100644
index 00000000..54fa29a3
--- /dev/null
+++ b/perllib/Graph/Attribute.pm
@@ -0,0 +1,130 @@
+package Graph::Attribute;
+
+use strict;
+
+sub _F () { 0 }
+sub _COMPAT02 () { 0x00000001 }
+
+sub import {
+    my $package = shift;
+    my %attr = @_;
+    my $caller = caller(0);
+    if (exists $attr{array}) {
+	my $i = $attr{array};
+	no strict 'refs';
+	*{"${caller}::_get_attributes"} = sub { $_[0]->[ $i ] };
+	*{"${caller}::_set_attributes"} =
+	    sub { $_[0]->[ $i ] ||= { };
+		  $_[0]->[ $i ] = $_[1] if @_ == 2;
+		  $_[0]->[ $i ] };
+	*{"${caller}::_has_attributes"} = sub { defined $_[0]->[ $i ] };
+	*{"${caller}::_delete_attributes"} = sub { undef $_[0]->[ $i ]; 1 };
+    } elsif (exists $attr{hash}) {
+	my $k = $attr{hash};
+	no strict 'refs';
+	*{"${caller}::_get_attributes"} = sub { $_[0]->{ $k } };
+	*{"${caller}::_set_attributes"} =
+	    sub { $_[0]->{ $k } ||= { };
+		  $_[0]->{ $k } = $_[1] if @_ == 2;
+		  $_[0]->{ $k } };
+	*{"${caller}::_has_attributes"} = sub { defined $_[0]->{ $k } };
+	*{"${caller}::_delete_attributes"} = sub { delete $_[0]->{ $k } };
+    } else {
+	die "Graph::Attribute::import($package @_) caller $caller\n";
+    }
+    my @api = qw(get_attribute
+		 get_attributes
+		 set_attribute
+		 set_attributes
+		 has_attribute
+		 has_attributes
+		 delete_attribute
+		 delete_attributes
+		 get_attribute_names
+		 get_attribute_values);
+    if (exists $attr{map}) {
+	my $map = $attr{map};
+	for my $api (@api) {
+	    my ($first, $rest) = ($api =~ /^(\w+?)_(.+)/);
+	    no strict 'refs';
+	    *{"${caller}::${first}_${map}_${rest}"} = \&$api;
+	}
+    }
+}
+
+sub set_attribute {
+    my $g = shift;
+    my $v = pop;
+    my $a = pop;
+    my $p = $g->_set_attributes;
+    $p->{ $a } = $v;
+    return 1;
+}
+
+sub set_attributes {
+    my $g = shift;
+    my $a = pop;
+    my $p = $g->_set_attributes( $a );
+    return 1;
+}
+
+sub has_attribute {
+    my $g = shift;
+    my $a = pop;
+    my $p = $g->_get_attributes;
+    $p ? exists $p->{ $a } : 0;
+}
+
+sub has_attributes {
+    my $g = shift;
+    $g->_get_attributes ? 1 : 0;
+}
+
+sub get_attribute {
+    my $g = shift;
+    my $a = pop;
+    my $p = $g->_get_attributes;
+    $p ? $p->{ $a } : undef;
+}
+
+sub delete_attribute {
+    my $g = shift;
+    my $a = pop;
+    my $p = $g->_get_attributes;
+    if (defined $p) {
+	delete $p->{ $a };
+	return 1;
+    } else {
+	return 0;
+    }
+}
+
+sub delete_attributes {
+    my $g = shift;
+    if ($g->_has_attributes) {
+	$g->_delete_attributes;
+	return 1;
+    } else {
+	return 0;
+    }
+}
+
+sub get_attribute_names {
+    my $g = shift;
+    my $p = $g->_get_attributes;
+    defined $p ? keys %{ $p } : ( );
+}
+
+sub get_attribute_values {
+    my $g = shift;
+    my $p = $g->_get_attributes;
+    defined $p ? values %{ $p } : ( );
+}
+
+sub get_attributes {
+    my $g = shift;
+    my $a = $g->_get_attributes;
+    ($g->[ _F ] & _COMPAT02) ? (defined $a ? %{ $a } : ()) : $a;
+}
+
+1;
diff --git a/perllib/Graph/BitMatrix.pm b/perllib/Graph/BitMatrix.pm
new file mode 100644
index 00000000..de913763
--- /dev/null
+++ b/perllib/Graph/BitMatrix.pm
@@ -0,0 +1,227 @@
+package Graph::BitMatrix;
+
+use strict;
+
+# $SIG{__DIE__ } = sub { use Carp; confess };
+# $SIG{__WARN__} = sub { use Carp; confess };
+
+sub _V () { 2 } # Graph::_V()
+sub _E () { 3 } # Graph::_E()
+sub _i () { 3 } # Index to path.
+sub _s () { 4 } # Successors / Path to Index.
+
+sub new {
+    my ($class, $g, %opt) = @_;
+    my @V = $g->vertices;
+    my $V = @V;
+    my $Z = "\0" x (($V + 7) / 8);
+    my %V; @V{ @V } = 0 .. $#V;
+    my $bm = bless [ [ ( $Z ) x $V ], \%V ], $class;
+    my $bm0 = $bm->[0];
+    my $connect_edges;
+    if (exists $opt{connect_edges}) {
+	$connect_edges = $opt{connect_edges};
+	delete $opt{connect_edges};
+    }
+    $connect_edges = 1 unless defined $connect_edges;
+    Graph::_opt_unknown(\%opt);
+    if ($connect_edges) {
+	# for (my $i = 0; $i <= $#V; $i++) {
+	#    my $u = $V[$i];
+	#    for (my $j = 0; $j <= $#V; $j++) {
+	#	vec($bm0->[$i], $j, 1) = 1 if $g->has_edge($u, $V[$j]);
+	#    }
+	# }
+	my $Vi = $g->[_V]->[_i];
+	my $Ei = $g->[_E]->[_i];
+	if ($g->is_undirected) {
+	    for my $e (keys %{ $Ei }) {
+		my ($i0, $j0) = @{ $Ei->{ $e } };
+		my $i1 = $V{ $Vi->{ $i0 } };
+		my $j1 = $V{ $Vi->{ $j0 } };
+		vec($bm0->[$i1], $j1, 1) = 1;
+		vec($bm0->[$j1], $i1, 1) = 1;
+	    }
+	} else {
+	    for my $e (keys %{ $Ei }) {
+		my ($i0, $j0) = @{ $Ei->{ $e } };
+		vec($bm0->[$V{ $Vi->{ $i0 } }], $V{ $Vi->{ $j0 } }, 1) = 1;
+	    }
+	}
+    }
+    return $bm;
+}
+
+sub set {
+    my ($m, $u, $v) = @_;
+    my ($i, $j) = map { $m->[1]->{ $_ } } ($u, $v);
+    vec($m->[0]->[$i], $j, 1) = 1 if defined $i && defined $j;
+}
+
+sub unset {
+    my ($m, $u, $v) = @_;
+    my ($i, $j) = map { $m->[1]->{ $_ } } ($u, $v);
+    vec($m->[0]->[$i], $j, 1) = 0 if defined $i && defined $j;
+}
+
+sub get {
+    my ($m, $u, $v) = @_;
+    my ($i, $j) = map { $m->[1]->{ $_ } } ($u, $v);
+    defined $i && defined $j ? vec($m->[0]->[$i], $j, 1) : undef;
+}
+
+sub set_row {
+    my ($m, $u) = splice @_, 0, 2;
+    my $m0 = $m->[0];
+    my $m1 = $m->[1];
+    my $i = $m1->{ $u };
+    return unless defined $i;
+    for my $v (@_) {
+	my $j = $m1->{ $v };
+	vec($m0->[$i], $j, 1) = 1 if defined $j;
+    }
+}
+
+sub unset_row {
+    my ($m, $u) = splice @_, 0, 2;
+    my $m0 = $m->[0];
+    my $m1 = $m->[1];
+    my $i = $m1->{ $u };
+    return unless defined $i;
+    for my $v (@_) {
+	my $j = $m1->{ $v };
+	vec($m0->[$i], $j, 1) = 0 if defined $j;
+    }
+}
+
+sub get_row {
+    my ($m, $u) = splice @_, 0, 2;
+    my $m0 = $m->[0];
+    my $m1 = $m->[1];
+    my $i = $m1->{ $u };
+    return () x @_ unless defined $i;
+    my @r;
+    for my $v (@_) {
+	my $j = $m1->{ $v };
+	push @r, defined $j ? (vec($m0->[$i], $j, 1) ? 1 : 0) : undef;
+    }
+    return @r;
+}
+
+sub vertices {
+    my ($m, $u, $v) = @_;
+    keys %{ $m->[1] };
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::BitMatrix - create and manipulate a V x V bit matrix of graph G
+
+=head1 SYNOPSIS
+
+    use Graph::BitMatrix;
+    use Graph::Directed;
+    my $g  = Graph::Directed->new;
+    $g->add_...(); # build $g
+    my $m = Graph::BitMatrix->new($g, %opt);
+    $m->get($u, $v)
+    $m->set($u, $v)
+    $m->unset($u, $v)
+    $m->get_row($u, $v1, $v2, ..., $vn)
+    $m->set_row($u, $v1, $v2, ..., $vn)
+    $m->unset_row($u, $v1, $v2, ..., $vn)
+    $a->vertices()
+
+=head1 DESCRIPTION
+
+This class enables creating bit matrices that compactly describe
+the connected of the graphs.
+
+=head2 Class Methods
+
+=over 4
+
+=item new($g)
+
+Create a bit matrix from a Graph $g.  The C<%opt>, if present,
+can have the following options:
+
+=over 8
+
+=item *
+
+connect_edges
+
+If true or if not present, set the bits in the bit matrix that
+correspond to edges.  If false, do not set any bits.  In either
+case the bit matrix of V x V bits is allocated.
+
+=back
+
+=back
+
+=head2 Object Methods
+
+=over 4
+
+=item get($u, $v)
+
+Return true if the bit matrix has a "one bit" between the vertices
+$u and $v; in other words, if there is (at least one) a vertex going from
+$u to $v.  If there is no vertex and therefore a "zero bit", return false.
+
+=item set($u, $v)
+
+Set the bit between the vertices $u and $v; in other words, connect
+the vertices $u and $v by an edge.  The change does not get mirrored
+back to the original graph.  Returns nothing.
+
+=item unset($u, $v)
+
+Unset the bit between the vertices $u and $v; in other words, disconnect
+the vertices $u and $v by an edge.  The change does not get mirrored
+back to the original graph.  Returns nothing.
+
+=item get_row($u, $v1, $v2, ..., $vn)
+
+Test the row at vertex C<u> for the vertices C<v1>, C<v2>, ..., C<vn>
+Returns a list of I<n> truth values.
+
+=item set_row($u, $v1, $v2, ..., $vn)
+
+Sets the row at vertex C<u> for the vertices C<v1>, C<v2>, ..., C<vn>,
+in other words, connects the vertex C<u> to the vertices C<vi>.
+The changes do not get mirrored back to the original graph.
+Returns nothing.
+
+=item unset_row($u, $v1, $v2, ..., $vn)
+
+Unsets the row at vertex C<u> for the vertices C<v1>, C<v2>, ..., C<vn>,
+in other words, disconnects the vertex C<u> from the vertices C<vi>.
+The changes do not get mirrored back to the original graph.
+Returns nothing.
+
+=item vertices
+
+Return the list of vertices in the bit matrix.
+
+=back
+
+=head1 ALGORITHM
+
+The algorithm used to create the matrix is two nested loops, which is
+O(V**2) in time, and the returned matrices are O(V**2) in space.
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/Directed.pm b/perllib/Graph/Directed.pm
new file mode 100644
index 00000000..9c3fc86d
--- /dev/null
+++ b/perllib/Graph/Directed.pm
@@ -0,0 +1,44 @@
+package Graph::Directed;
+
+use Graph;
+use base 'Graph';
+use strict;
+
+=pod
+
+=head1 NAME
+
+Graph::Directed - directed graphs
+
+=head1 SYNOPSIS
+
+    use Graph::Directed;
+    my $g = Graph::Directed->new;
+
+    # Or alternatively:
+
+    use Graph;
+    my $g = Graph->new(directed => 1);
+    my $g = Graph->new(undirected => 0);
+
+=head1 DESCRIPTION
+
+Graph::Directed allows you to create directed graphs.
+
+For the available methods, see L<Graph>.
+
+=head1 SEE ALSO
+
+L<Graph>, L<Graph::Undirected>
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
+
+1;
diff --git a/perllib/Graph/MSTHeapElem.pm b/perllib/Graph/MSTHeapElem.pm
new file mode 100644
index 00000000..32bc0011
--- /dev/null
+++ b/perllib/Graph/MSTHeapElem.pm
@@ -0,0 +1,24 @@
+package Graph::MSTHeapElem;
+
+use strict;
+use vars qw($VERSION @ISA);
+use Heap071::Elem;
+
+use base 'Heap071::Elem';
+
+$VERSION = 0.01;
+
+sub new {
+    my $class = shift;
+    bless { u => $_[0], v => $_[1], w => $_[2] }, $class;
+}
+
+sub cmp {
+    ($_[0]->{ w } || 0) <=> ($_[1]->{ w } || 0);
+}
+
+sub val {
+    @{ $_[0] }{ qw(u v w) };
+}
+
+1;
diff --git a/perllib/Graph/Matrix.pm b/perllib/Graph/Matrix.pm
new file mode 100644
index 00000000..d3b9d407
--- /dev/null
+++ b/perllib/Graph/Matrix.pm
@@ -0,0 +1,82 @@
+package Graph::Matrix;
+
+# $SIG{__DIE__ } = sub { use Carp; confess };
+# $SIG{__WARN__} = sub { use Carp; confess };
+
+use strict;
+
+sub new {
+    my ($class, $g) = @_;
+    my @V = $g->vertices;
+    my $V = @V;
+    my %V; @V{ @V } = 0 .. $#V;
+    bless [ [ map { [ ] } 0 .. $#V ], \%V ], $class;
+}
+
+sub set {
+    my ($m, $u, $v, $val) = @_;
+    my ($i, $j) = map { $m->[1]->{ $_ } } ($u, $v);
+    $m->[0]->[$i]->[$j] = $val;
+}
+
+sub get {
+    my ($m, $u, $v) = @_;
+    my ($i, $j) = map { $m->[1]->{ $_ } } ($u, $v);
+    $m->[0]->[$i]->[$j];
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::Matrix - create and manipulate a V x V matrix of graph G
+
+=head1 SYNOPSIS
+
+    use Graph::Matrix;
+    use Graph::Directed;
+    my $g  = Graph::Directed->new;
+    $g->add_...(); # build $g
+    my $m = Graph::Matrix->new($g);
+    $m->get($u, $v)
+    $s->get($u, $v, $val)
+
+=head1 DESCRIPTION
+
+B<This module is meant for internal use by the Graph module.>
+
+=head2 Class Methods
+
+=over 4
+
+=item new($g)
+
+Construct a new Matrix from the Graph $g.
+
+=back
+
+=head2 Object Methods
+
+=over 4
+
+=item get($u, $v)
+
+Return the value at the edge from $u to $v.
+
+=item set($u, $v, $val)
+
+Set the edge from $u to $v to value $val.
+
+=back
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/SPTHeapElem.pm b/perllib/Graph/SPTHeapElem.pm
new file mode 100644
index 00000000..04555310
--- /dev/null
+++ b/perllib/Graph/SPTHeapElem.pm
@@ -0,0 +1,26 @@
+package Graph::SPTHeapElem;
+
+use strict;
+use vars qw($VERSION @ISA);
+use Heap071::Elem;
+
+use base 'Heap071::Elem';
+
+$VERSION = 0.01;
+
+sub new {
+    my $class = shift;
+    bless { u => $_[0], v => $_[1], w => $_[2] }, $class;
+}
+
+sub cmp {
+    ($_[0]->{ w } || 0) <=> ($_[1]->{ w } || 0) ||
+    ($_[0]->{ u } cmp $_[1]->{ u }) ||
+    ($_[0]->{ u } cmp $_[1]->{ v });
+}
+
+sub val {
+    @{ $_[0] }{ qw(u v w) };
+}
+
+1;
diff --git a/perllib/Graph/TransitiveClosure.pm b/perllib/Graph/TransitiveClosure.pm
new file mode 100644
index 00000000..fd5a0a82
--- /dev/null
+++ b/perllib/Graph/TransitiveClosure.pm
@@ -0,0 +1,155 @@
+package Graph::TransitiveClosure;
+
+# COMMENT THESE OUT FOR TESTING AND PRODUCTION.
+# $SIG{__DIE__ } = sub { use Carp; confess };
+# $SIG{__WARN__} = sub { use Carp; confess };
+
+use base 'Graph';
+use Graph::TransitiveClosure::Matrix;
+
+sub _G () { Graph::_G() }
+
+sub new {
+    my ($class, $g, %opt) = @_;
+    $g->expect_non_multiedged;
+    %opt = (path_vertices => 1) unless %opt;
+    my $attr = Graph::_defattr();
+    if (exists $opt{ attribute_name }) {
+	$attr = $opt{ attribute_name };
+	# No delete $opt{ attribute_name } since we need to pass it on.
+    }
+    $opt{ reflexive } = 1 unless exists $opt{ reflexive };
+    my $tcm = $g->new( $opt{ reflexive } ?
+		       ( vertices => [ $g->vertices ] ) : ( ) );
+    my $tcg = $g->get_graph_attribute('_tcg');
+    if (defined $tcg && $tcg->[ 0 ] == $g->[ _G ]) {
+	$tcg = $tcg->[ 1 ];
+    } else {
+	$tcg = Graph::TransitiveClosure::Matrix->new($g, %opt);
+	$g->set_graph_attribute('_tcg', [ $g->[ _G ], $tcg ]);
+    }
+    my $tcg00 = $tcg->[0]->[0];
+    my $tcg11 = $tcg->[1]->[1];
+    for my $u ($tcg->vertices) {
+	my $tcg00i = $tcg00->[ $tcg11->{ $u } ];
+	for my $v ($tcg->vertices) {
+	    next if $u eq $v && ! $opt{ reflexive };
+	    my $j = $tcg11->{ $v };
+	    if (
+		# $tcg->is_transitive($u, $v)
+		# $tcg->[0]->get($u, $v)
+		vec($tcg00i, $j, 1)
+	       ) {
+		my $val = $g->_get_edge_attribute($u, $v, $attr);
+		$tcm->_set_edge_attribute($u, $v, $attr,
+					  defined $val ? $val :
+					  $u eq $v ?
+					  0 : 1);
+	    }
+	}
+    }
+    $tcm->set_graph_attribute('_tcm', $tcg);
+    bless $tcm, $class;
+}
+
+sub is_transitive {
+    my $g = shift;
+    Graph::TransitiveClosure::Matrix::is_transitive($g);
+}
+
+1;
+__END__
+=pod
+
+Graph::TransitiveClosure - create and query transitive closure of graph
+
+=head1 SYNOPSIS
+
+    use Graph::TransitiveClosure;
+    use Graph::Directed; # or Undirected
+
+    my $g  = Graph::Directed->new;
+    $g->add_...(); # build $g
+
+    # Compute the transitive closure graph.
+    my $tcg = Graph::TransitiveClosure->new($g);
+    $tcg->is_reachable($u, $v) # Identical to $tcg->has_edge($u, $v)
+
+    # Being reflexive is the default, meaning that null transitions
+    # (transitions from a vertex to the same vertex) are included.
+    my $tcg = Graph::TransitiveClosure->new($g, reflexive => 1);
+    my $tcg = Graph::TransitiveClosure->new($g, reflexive => 0);
+
+    # is_reachable(u, v) is always reflexive.
+    $tcg->is_reachable($u, $v)
+
+    # The reflexivity of is_transitive(u, v) depends of the reflexivity
+    # of the transitive closure.
+    $tcg->is_transitive($u, $v)
+
+    # You can check any graph for transitivity.
+    $g->is_transitive()
+
+    my $tcg = Graph::TransitiveClosure->new($g, path_length => 1);
+    $tcg->path_length($u, $v)
+
+    # path_vertices is automatically always on so this is a no-op.
+    my $tcg = Graph::TransitiveClosure->new($g, path_vertices => 1);
+    $tcg->path_vertices($u, $v)
+
+    # Both path_length and path_vertices.
+    my $tcg = Graph::TransitiveClosure->new($g, path => 1);
+    $tcg->path_vertices($u, $v)
+    $tcg->length($u, $v)
+
+    my $tcg = Graph::TransitiveClosure->new($g, attribute_name => 'length');
+    $tcg->path_length($u, $v)
+
+=head1 DESCRIPTION
+
+You can use C<Graph::TransitiveClosure> to compute the transitive
+closure graph of a graph and optionally also the minimum paths
+(lengths and vertices) between vertices, and after that query the
+transitiveness between vertices by using the C<is_reachable()> and
+C<is_transitive()> methods, and the paths by using the
+C<path_length()> and C<path_vertices()> methods.
+
+For further documentation, see the L<Graph::TransitiveClosure::Matrix>.
+
+=head2 Class Methods
+
+=over 4
+
+=item new($g, %opt)
+
+Construct a new transitive closure object.  Note that strictly speaking
+the returned object is not a graph; it is a graph plus other stuff.  But
+you should be able to use it as a graph plus a couple of methods inherited
+from the Graph::TransitiveClosure::Matrix class.
+
+=back
+
+=head2 Object Methods
+
+These are only the methods 'native' to the class: see
+L<Graph::TransitiveClosure::Matrix> for more.
+
+=over 4
+
+=item is_transitive($g)
+
+Return true if the Graph $g is transitive.
+
+=item transitive_closure_matrix
+
+Return the transitive closure matrix of the transitive closure object.
+
+=back
+
+=head2 INTERNALS
+
+The transitive closure matrix is stored as an attribute of the graph
+called C<_tcm>, and any methods not found in the graph class are searched
+in the transitive closure matrix class. 
+
+=cut
diff --git a/perllib/Graph/TransitiveClosure/Matrix.pm b/perllib/Graph/TransitiveClosure/Matrix.pm
new file mode 100644
index 00000000..be56f2a9
--- /dev/null
+++ b/perllib/Graph/TransitiveClosure/Matrix.pm
@@ -0,0 +1,488 @@
+package Graph::TransitiveClosure::Matrix;
+
+use strict;
+
+use Graph::AdjacencyMatrix;
+use Graph::Matrix;
+
+sub _new {
+    my ($g, $class, $opt, $want_transitive, $want_reflexive, $want_path, $want_path_vertices) = @_;
+    my $m = Graph::AdjacencyMatrix->new($g, %$opt);
+    my @V = $g->vertices;
+    my $am = $m->adjacency_matrix;
+    my $dm; # The distance matrix.
+    my $pm; # The predecessor matrix.
+    my @di;
+    my %di; @di{ @V } = 0..$#V;
+    my @ai = @{ $am->[0] };
+    my %ai = %{ $am->[1] };
+    my @pi;
+    my %pi;
+    unless ($want_transitive) {
+	$dm = $m->distance_matrix;
+	@di = @{ $dm->[0] };
+	%di = %{ $dm->[1] };
+	$pm = Graph::Matrix->new($g);
+	@pi = @{ $pm->[0] };
+	%pi = %{ $pm->[1] };
+	for my $u (@V) {
+	    my $diu = $di{$u};
+	    my $aiu = $ai{$u};
+	    for my $v (@V) {
+		my $div = $di{$v};
+		my $aiv = $ai{$v};
+		next unless
+		    # $am->get($u, $v)
+		    vec($ai[$aiu], $aiv, 1)
+			;
+		# $dm->set($u, $v, $u eq $v ? 0 : 1)
+		$di[$diu]->[$div] = $u eq $v ? 0 : 1
+		    unless
+			defined
+			    # $dm->get($u, $v)
+			    $di[$diu]->[$div]
+			    ;
+		$pi[$diu]->[$div] = $v unless $u eq $v;
+	    }
+	}
+    }
+    # XXX (see the bits below): sometimes, being nice and clean is the
+    # wrong thing to do.  In this case, using the public API for graph
+    # transitive matrices and bitmatrices makes things awfully slow.
+    # Instead, we go straight for the jugular of the data structures.
+    for my $u (@V) {
+	my $diu = $di{$u};
+	my $aiu = $ai{$u};
+	my $didiu = $di[$diu];
+	my $aiaiu = $ai[$aiu];
+	for my $v (@V) {
+	    my $div = $di{$v};
+	    my $aiv = $ai{$v};
+	    my $didiv = $di[$div];
+	    my $aiaiv = $ai[$aiv];
+	    if (
+		# $am->get($v, $u)
+		vec($aiaiv, $aiu, 1)
+		|| ($want_reflexive && $u eq $v)) {
+		my $aivivo = $aiaiv;
+		if ($want_transitive) {
+		    if ($want_reflexive) {
+			for my $w (@V) {
+			    next if $w eq $u;
+			    my $aiw = $ai{$w};
+			    return 0
+				if  vec($aiaiu, $aiw, 1) &&
+				   !vec($aiaiv, $aiw, 1);
+			}
+			# See XXX above.
+			# for my $w (@V) {
+			#    my $aiw = $ai{$w};
+			#    if (
+			#	# $am->get($u, $w)
+			#	vec($aiaiu, $aiw, 1)
+			#	|| ($u eq $w)) {
+			#	return 0
+			#	    if $u ne $w &&
+			#		# !$am->get($v, $w)
+			#		!vec($aiaiv, $aiw, 1)
+			#		    ;
+			#	# $am->set($v, $w)
+			#	vec($aiaiv, $aiw, 1) = 1
+			#	    ;
+			#     }
+			# }
+		    } else {
+			# See XXX above.
+			# for my $w (@V) {
+			#     my $aiw = $ai{$w};
+			#     if (
+			#	# $am->get($u, $w)
+			#	vec($aiaiu, $aiw, 1)
+			#       ) {
+			#	return 0
+			#	    if $u ne $w &&
+			#		# !$am->get($v, $w)
+			#		!vec($aiaiv, $aiw, 1)
+			#		    ;
+			# 	# $am->set($v, $w)
+			# 	vec($aiaiv, $aiw, 1) = 1
+			# 	    ;
+			#     }
+			# }
+			$aiaiv |= $aiaiu;
+		    }
+		} else {
+		    if ($want_reflexive) {
+			$aiaiv |= $aiaiu;
+			vec($aiaiv, $aiu, 1) = 1;
+			# See XXX above.
+			# for my $w (@V) {
+			#     my $aiw = $ai{$w};
+			#     if (
+			# 	# $am->get($u, $w)
+			#	vec($aiaiu, $aiw, 1)
+			#	|| ($u eq $w)) {
+			#	# $am->set($v, $w)
+			#	vec($aiaiv, $aiw, 1) = 1
+			#	    ;
+			#     }
+			# }
+		    } else {
+			$aiaiv |= $aiaiu;
+			# See XXX above.
+			# for my $w (@V) {
+			#    my $aiw = $ai{$w};
+			#    if (
+			#	# $am->get($u, $w)
+			#	vec($aiaiu, $aiw, 1)
+			#       ) {
+			#	# $am->set($v, $w)
+			#	vec($aiaiv, $aiw, 1) = 1
+			#	    ;
+			#     }
+			# }
+		    }
+		}
+		if ($aiaiv ne $aivivo) {
+		    $ai[$aiv] = $aiaiv;
+		    $aiaiu = $aiaiv if $u eq $v;
+		}
+	    }
+	    if ($want_path && !$want_transitive) {
+		for my $w (@V) {
+		    my $aiw = $ai{$w};
+		    next unless
+			# See XXX above.
+			# $am->get($v, $u)
+			vec($aiaiv, $aiu, 1)
+			    &&
+			# See XXX above.
+			# $am->get($u, $w)
+			vec($aiaiu, $aiw, 1)
+			    ;
+		    my $diw = $di{$w};
+		    my ($d0, $d1a, $d1b);
+		    if (defined $dm) {
+			# See XXX above.
+			# $d0  = $dm->get($v, $w);
+			# $d1a = $dm->get($v, $u) || 1;
+			# $d1b = $dm->get($u, $w) || 1;
+			$d0  = $didiv->[$diw];
+			$d1a = $didiv->[$diu] || 1;
+			$d1b = $didiu->[$diw] || 1;
+		    } else {
+			$d1a = 1;
+			$d1b = 1;
+		    }
+		    my $d1 = $d1a + $d1b;
+		    if (!defined $d0 || ($d1 < $d0)) {
+			# print "d1 = $d1a ($v, $u) + $d1b ($u, $w) = $d1 ($v, $w) (".(defined$d0?$d0:"-").")\n";
+			# See XXX above.
+			# $dm->set($v, $w, $d1);
+			$didiv->[$diw] = $d1;
+			$pi[$div]->[$diw] = $pi[$div]->[$diu]
+			    if $want_path_vertices;
+		    }
+		}
+		# $dm->set($u, $v, 1)
+		$didiu->[$div] = 1
+		    if $u ne $v &&
+		       # $am->get($u, $v)
+		       vec($aiaiu, $aiv, 1)
+			   &&
+		       # !defined $dm->get($u, $v);
+		       !defined $didiu->[$div];
+	    }
+	}
+    }
+    return 1 if $want_transitive;
+    my %V; @V{ @V } = @V;
+    $am->[0] = \@ai;
+    $am->[1] = \%ai;
+    if (defined $dm) {
+	$dm->[0] = \@di;
+	$dm->[1] = \%di;
+    }
+    if (defined $pm) {
+	$pm->[0] = \@pi;
+	$pm->[1] = \%pi;
+    }
+    bless [ $am, $dm, $pm, \%V ], $class;
+}
+
+sub new {
+    my ($class, $g, %opt) = @_;
+    my %am_opt = (distance_matrix => 1);
+    if (exists $opt{attribute_name}) {
+	$am_opt{attribute_name} = $opt{attribute_name};
+	delete $opt{attribute_name};
+    }
+    if ($opt{distance_matrix}) {
+	$am_opt{distance_matrix} = $opt{distance_matrix};
+    }
+    delete $opt{distance_matrix};
+    if (exists $opt{path}) {
+	$opt{path_length}   = $opt{path};
+	$opt{path_vertices} = $opt{path};
+	delete $opt{path};
+    }
+    my $want_path_length;
+    if (exists $opt{path_length}) {
+	$want_path_length = $opt{path_length};
+	delete $opt{path_length};
+    }
+    my $want_path_vertices;
+    if (exists $opt{path_vertices}) {
+	$want_path_vertices = $opt{path_vertices};
+	delete $opt{path_vertices};
+    }
+    my $want_reflexive;
+    if (exists $opt{reflexive}) {
+	$want_reflexive = $opt{reflexive};
+	delete $opt{reflexive};
+    }
+    my $want_transitive;
+    if (exists $opt{is_transitive}) {
+	$want_transitive = $opt{is_transitive};
+	$am_opt{is_transitive} = $want_transitive;
+	delete $opt{is_transitive};
+    }
+    die "Graph::TransitiveClosure::Matrix::new: Unknown options: @{[map { qq['$_' => $opt{$_}]} keys %opt]}"
+	if keys %opt;
+    $want_reflexive = 1 unless defined $want_reflexive;
+    my $want_path = $want_path_length || $want_path_vertices;
+    # $g->expect_dag if $want_path;
+    _new($g, $class,
+	 \%am_opt,
+	 $want_transitive, $want_reflexive,
+	 $want_path, $want_path_vertices);
+}
+
+sub has_vertices {
+    my $tc = shift;
+    for my $v (@_) {
+	return 0 unless exists $tc->[3]->{ $v };
+    }
+    return 1;
+}
+
+sub is_reachable {
+    my ($tc, $u, $v) = @_;
+    return undef unless $tc->has_vertices($u, $v);
+    return 1 if $u eq $v;
+    $tc->[0]->get($u, $v);
+}
+
+sub is_transitive {
+    if (@_ == 1) {	# Any graph.
+	__PACKAGE__->new($_[0], is_transitive => 1);	# Scary.
+    } else {		# A TC graph.
+	my ($tc, $u, $v) = @_;
+	return undef unless $tc->has_vertices($u, $v);
+	$tc->[0]->get($u, $v);
+    }
+}
+
+sub vertices {
+    my $tc = shift;
+    values %{ $tc->[3] };
+}
+
+sub path_length {
+    my ($tc, $u, $v) = @_;
+    return undef unless $tc->has_vertices($u, $v);
+    return 0 if $u eq $v;
+    $tc->[1]->get($u, $v);
+}
+
+sub path_predecessor {
+    my ($tc, $u, $v) = @_;
+    return undef if $u eq $v;
+    return undef unless $tc->has_vertices($u, $v);
+    $tc->[2]->get($u, $v);
+}
+
+sub path_vertices {
+    my ($tc, $u, $v) = @_;
+    return unless $tc->is_reachable($u, $v);
+    return wantarray ? () : 0 if $u eq $v;
+    my @v = ( $u );
+    while ($u ne $v) {
+	last unless defined($u = $tc->path_predecessor($u, $v));
+	push @v, $u;
+    }
+    $tc->[2]->set($u, $v, [ @v ]) if @v;
+    return @v;
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::TransitiveClosure::Matrix - create and query transitive closure of graph
+
+=head1 SYNOPSIS
+
+    use Graph::TransitiveClosure::Matrix;
+    use Graph::Directed; # or Undirected
+
+    my $g  = Graph::Directed->new;
+    $g->add_...(); # build $g
+
+    # Compute the transitive closure matrix.
+    my $tcm = Graph::TransitiveClosure::Matrix->new($g);
+
+    # Being reflexive is the default,
+    # meaning that null transitions are included.
+    my $tcm = Graph::TransitiveClosure::Matrix->new($g, reflexive => 1);
+    $tcm->is_reachable($u, $v)
+
+    # is_reachable(u, v) is always reflexive.
+    $tcm->is_reachable($u, $v)
+
+    # The reflexivity of is_transitive(u, v) depends of the reflexivity
+    # of the transitive closure.
+    $tcg->is_transitive($u, $v)
+
+    my $tcm = Graph::TransitiveClosure::Matrix->new($g, path_length => 1);
+    $tcm->path_length($u, $v)
+
+    my $tcm = Graph::TransitiveClosure::Matrix->new($g, path_vertices => 1);
+    $tcm->path_vertices($u, $v)
+
+    my $tcm = Graph::TransitiveClosure::Matrix->new($g, attribute_name => 'length');
+    $tcm->path_length($u, $v)
+
+    $tcm->vertices
+
+=head1 DESCRIPTION
+
+You can use C<Graph::TransitiveClosure::Matrix> to compute the
+transitive closure matrix of a graph and optionally also the minimum
+paths (lengths and vertices) between vertices, and after that query
+the transitiveness between vertices by using the C<is_reachable()> and
+C<is_transitive()> methods, and the paths by using the
+C<path_length()> and C<path_vertices()> methods.
+
+If you modify the graph after computing its transitive closure,
+the transitive closure and minimum paths may become invalid.
+
+=head1 Methods
+
+=head2 Class Methods
+
+=over 4
+
+=item new($g)
+
+Construct the transitive closure matrix of the graph $g.
+
+=item new($g, options)
+
+Construct the transitive closure matrix of the graph $g with options
+as a hash. The known options are
+
+=over 8
+
+=item C<attribute_name> => I<attribute_name>
+
+By default the edge attribute used for distance is C<w>.  You can
+change that by giving another attribute name with the C<attribute_name>
+attribute to the new() constructor.
+
+=item reflexive => boolean
+
+By default the transitive closure matrix is not reflexive: that is,
+the adjacency matrix has zeroes on the diagonal.  To have ones on
+the diagonal, use true for the C<reflexive> option.
+
+B<NOTE>: this behaviour has changed from Graph 0.2xxx: transitive
+closure graphs were by default reflexive.
+
+=item path_length => boolean
+
+By default the path lengths are not computed, only the boolean transitivity.
+By using true for C<path_length> also the path lengths will be computed,
+they can be retrieved using the path_length() method.
+
+=item path_vertices => boolean
+
+By default the paths are not computed, only the boolean transitivity.
+By using true for C<path_vertices> also the paths will be computed,
+they can be retrieved using the path_vertices() method.
+
+=back
+
+=back
+
+=head2 Object Methods
+
+=over 4
+
+=item is_reachable($u, $v)
+
+Return true if the vertex $v is reachable from the vertex $u,
+or false if not.
+
+=item path_length($u, $v)
+
+Return the minimum path length from the vertex $u to the vertex $v,
+or undef if there is no such path.
+
+=item path_vertices($u, $v)
+
+Return the minimum path (as a list of vertices) from the vertex $u to
+the vertex $v, or an empty list if there is no such path, OR also return
+an empty list if $u equals $v.
+
+=item has_vertices($u, $v, ...)
+
+Return true if the transitive closure matrix has all the listed vertices,
+false if not.
+
+=item is_transitive($u, $v)
+
+Return true if the vertex $v is transitively reachable from the vertex $u,
+false if not.
+
+=item vertices
+
+Return the list of vertices in the transitive closure matrix.
+
+=item path_predecessor
+
+Return the predecessor of vertex $v in the transitive closure path
+going back to vertex $u.
+
+=back
+
+=head1 RETURN VALUES
+
+For path_length() the return value will be the sum of the appropriate
+attributes on the edges of the path, C<weight> by default.  If no
+attribute has been set, one (1) will be assumed.
+
+If you try to ask about vertices not in the graph, undefs and empty
+lists will be returned.
+
+=head1 ALGORITHM
+
+The transitive closure algorithm used is Warshall and Floyd-Warshall
+for the minimum paths, which is O(V**3) in time, and the returned
+matrices are O(V**2) in space.
+
+=head1 SEE ALSO
+
+L<Graph::AdjacencyMatrix>
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/Traversal.pm b/perllib/Graph/Traversal.pm
new file mode 100644
index 00000000..edfc5b19
--- /dev/null
+++ b/perllib/Graph/Traversal.pm
@@ -0,0 +1,714 @@
+package Graph::Traversal;
+
+use strict;
+
+# $SIG{__DIE__ } = sub { use Carp; confess };
+# $SIG{__WARN__} = sub { use Carp; confess };
+
+sub DEBUG () { 0 }
+
+sub reset {
+    my $self = shift;
+    $self->{ unseen } = { map { $_ => $_ } $self->{ graph }->vertices };
+    $self->{ seen   } = { };
+    $self->{ order     } = [ ];
+    $self->{ preorder  } = [ ];
+    $self->{ postorder } = [ ];
+    $self->{ roots     } = [ ];
+    $self->{ tree      } =
+	Graph->new( directed => $self->{ graph }->directed );
+    delete $self->{ terminate };
+}
+
+my $see = sub {
+    my $self = shift;
+    $self->see;
+};
+
+my $see_active = sub {
+    my $self = shift;
+    delete @{ $self->{ active } }{ $self->see };
+};
+
+sub has_a_cycle {
+    my ($u, $v, $t, $s) = @_;
+    $s->{ has_a_cycle } = 1;
+    $t->terminate;
+}
+
+sub find_a_cycle {
+    my ($u, $v, $t, $s) = @_;
+    my @cycle = ( $u );
+    push @cycle, $v unless $u eq $v;
+    my $path  = $t->{ order };
+    if (@$path) {
+	my $i = $#$path;
+	while ($i >= 0 && $path->[ $i ] ne $v) { $i-- }
+	if ($i >= 0) {
+	    unshift @cycle, @{ $path }[ $i+1 .. $#$path ];
+	}
+    }
+    $s->{ a_cycle } = \@cycle;
+    $t->terminate;
+}
+
+sub configure {
+    my ($self, %attr) = @_;
+    $self->{ pre  } = $attr{ pre }  if exists $attr{ pre  };
+    $self->{ post } = $attr{ post } if exists $attr{ post };
+    $self->{ pre_vertex  } = $attr{ pre_vertex }  if exists $attr{ pre_vertex  };
+    $self->{ post_vertex } = $attr{ post_vertex } if exists $attr{ post_vertex };
+    $self->{ pre_edge  } = $attr{ pre_edge  } if exists $attr{ pre_edge  };
+    $self->{ post_edge } = $attr{ post_edge } if exists $attr{ post_edge };
+    if (exists $attr{ successor }) { # Graph 0.201 compatibility.
+	$self->{ tree_edge } = $self->{ non_tree_edge } = $attr{ successor };
+    }
+    if (exists $attr{ unseen_successor }) {
+	if (exists $self->{ tree_edge }) { # Graph 0.201 compatibility.
+	    my $old_tree_edge = $self->{ tree_edge };
+	    $self->{ tree_edge } = sub {
+		$old_tree_edge->( @_ );
+		$attr{ unseen_successor }->( @_ );
+	    };
+	} else {
+	    $self->{ tree_edge } = $attr{ unseen_successor };
+	}
+    }
+    if ($self->graph->multiedged || $self->graph->countedged) {
+	$self->{ seen_edge } = $attr{ seen_edge } if exists $attr{ seen_edge };
+	if (exists $attr{ seen_successor }) { # Graph 0.201 compatibility.
+	    $self->{ seen_edge } = $attr{ seen_edge };
+	}
+    }
+    $self->{ non_tree_edge } = $attr{ non_tree_edge } if exists $attr{ non_tree_edge };
+    $self->{ pre_edge  } = $attr{ tree_edge } if exists $attr{ tree_edge };
+    $self->{ back_edge } = $attr{ back_edge } if exists $attr{ back_edge };
+    $self->{ down_edge } = $attr{ down_edge } if exists $attr{ down_edge };
+    $self->{ cross_edge } = $attr{ cross_edge } if exists $attr{ cross_edge };
+    if (exists $attr{ start }) {
+	$attr{ first_root } = $attr{ start };
+	$attr{ next_root  } = undef;
+    }
+    if (exists $attr{ get_next_root }) {
+	$attr{ next_root  } = $attr{ get_next_root }; # Graph 0.201 compat.
+    }
+    $self->{ next_root } =
+	exists $attr{ next_root } ?
+	    $attr{ next_root } :
+		$attr{ next_alphabetic } ?
+		    \&Graph::_next_alphabetic :
+			$attr{ next_numeric } ?
+			    \&Graph::_next_numeric :
+				\&Graph::_next_random;
+    $self->{ first_root } =
+	exists $attr{ first_root } ?
+	    $attr{ first_root } :
+		exists $attr{ next_root } ?
+		    $attr{ next_root } :
+			$attr{ next_alphabetic } ?
+			    \&Graph::_next_alphabetic :
+				$attr{ next_numeric } ?
+				    \&Graph::_next_numeric :
+					\&Graph::_next_random;
+    $self->{ next_successor } =
+	exists $attr{ next_successor } ?
+	    $attr{ next_successor } :
+		$attr{ next_alphabetic } ?
+		    \&Graph::_next_alphabetic :
+			$attr{ next_numeric } ?
+			    \&Graph::_next_numeric :
+				\&Graph::_next_random;
+    if (exists $attr{ has_a_cycle }) {
+	my $has_a_cycle =
+	    ref $attr{ has_a_cycle } eq 'CODE' ?
+		$attr{ has_a_cycle } : \&has_a_cycle;
+	$self->{ back_edge } = $has_a_cycle;
+	if ($self->{ graph }->is_undirected) {
+	    $self->{ down_edge } = $has_a_cycle;
+	}
+    }
+    if (exists $attr{ find_a_cycle }) {
+	my $find_a_cycle =
+	    ref $attr{ find_a_cycle } eq 'CODE' ?
+		$attr{ find_a_cycle } : \&find_a_cycle;
+	$self->{ back_edge } = $find_a_cycle;
+	if ($self->{ graph }->is_undirected) {
+	    $self->{ down_edge } = $find_a_cycle;
+	}
+    }
+    $self->{ add } = \&add_order;
+    $self->{ see } = $see;
+    delete @attr{ qw(
+		     pre post pre_edge post_edge
+		     successor unseen_successor seen_successor
+		     tree_edge non_tree_edge
+		     back_edge down_edge cross_edge seen_edge
+		     start get_next_root
+		     next_root next_alphabetic next_numeric next_random next_successor
+		     first_root
+		     has_a_cycle find_a_cycle
+		    ) };
+    if (keys %attr) {
+	require Carp;
+	my @attr = sort keys %attr;
+	Carp::croak(sprintf "Graph::Traversal: unknown attribute%s @{[map { qq['$_'] } @attr]}\n", @attr == 1 ? '' : 's');
+    }
+}
+
+sub new {
+    my $class = shift;
+    my $g = shift;
+    unless (ref $g && $g->isa('Graph')) {
+	require Carp;
+	Carp::croak("Graph::Traversal: first argument is not a Graph");
+    }
+    my $self = { graph => $g, state => { } };
+    bless $self, $class;
+    $self->reset;
+    $self->configure( @_ );
+    return $self;
+}
+
+sub terminate {
+    my $self = shift;
+    $self->{ terminate } = 1;
+}
+
+sub add_order {
+    my ($self, @next) = @_;
+    push @{ $self->{ order } }, @next;
+}
+
+sub visit {
+    my ($self, @next) = @_;
+    delete @{ $self->{ unseen } }{ @next };
+    print "unseen = @{[sort keys %{$self->{unseen}}]}\n" if DEBUG;
+    @{ $self->{ seen } }{ @next } = @next;
+    print "seen = @{[sort keys %{$self->{seen}}]}\n" if DEBUG;
+    $self->{ add }->( $self, @next );
+    print "order = @{$self->{order}}\n" if DEBUG;
+    if (exists $self->{ pre }) {
+	my $p = $self->{ pre };
+	for my $v (@next) {
+	    $p->( $v, $self );
+	}
+    }
+}
+
+sub visit_preorder {
+    my ($self, @next) = @_;
+    push @{ $self->{ preorder } }, @next;
+    for my $v (@next) {
+	$self->{ preordern }->{ $v } = $self->{ preorderi }++;
+    }
+    print "preorder = @{$self->{preorder}}\n" if DEBUG;
+    $self->visit( @next );
+}
+
+sub visit_postorder {
+    my ($self) = @_;
+    my @post = reverse $self->{ see }->( $self );
+    push @{ $self->{ postorder } }, @post;
+    for my $v (@post) {
+	$self->{ postordern }->{ $v } = $self->{ postorderi }++;
+    }
+    print "postorder = @{$self->{postorder}}\n" if DEBUG;
+    if (exists $self->{ post }) {
+	my $p = $self->{ post };
+	for my $v (@post) {
+	    $p->( $v, $self ) ;
+	}
+    }
+    if (exists $self->{ post_edge }) {
+	my $p = $self->{ post_edge };
+	my $u = $self->current;
+	if (defined $u) {
+	    for my $v (@post) {
+		$p->( $u, $v, $self, $self->{ state });
+	    }
+	}
+    }
+}
+
+sub _callbacks {
+    my ($self, $current, @all) = @_;
+    return unless @all;
+    my $nontree  = $self->{ non_tree_edge };
+    my $back     = $self->{ back_edge };
+    my $down     = $self->{ down_edge };
+    my $cross    = $self->{ cross_edge };
+    my $seen     = $self->{ seen_edge };
+    my $bdc = defined $back || defined $down || defined $cross;
+    if (defined $nontree || $bdc || defined $seen) {
+	my $u = $current;
+	my $preu  = $self->{ preordern  }->{ $u };
+	my $postu = $self->{ postordern }->{ $u };
+	for my $v ( @all ) {
+	    my $e = $self->{ tree }->has_edge( $u, $v );
+	    if ( !$e && (defined $nontree || $bdc) ) {
+		if ( exists $self->{ seen }->{ $v }) {
+		    $nontree->( $u, $v, $self, $self->{ state })
+			if $nontree;
+		    if ($bdc) {
+			my $postv = $self->{ postordern }->{ $v };
+			if ($back &&
+			    (!defined $postv || $postv >= $postu)) {
+			    $back ->( $u, $v, $self, $self->{ state });
+			} else {
+			    my $prev = $self->{ preordern }->{ $v };
+			    if ($down && $prev > $preu) {
+				$down ->( $u, $v, $self, $self->{ state });
+			    } elsif ($cross && $prev < $preu) {
+				$cross->( $u, $v, $self, $self->{ state });
+			    }
+			}
+		    }
+		}
+	    }
+	    if ($seen) {
+		my $c = $self->graph->get_edge_count($u, $v);
+		while ($c-- > 1) {
+		    $seen->( $u, $v, $self, $self->{ state } );
+		}
+	    }
+	}
+    }
+}
+
+sub next {
+    my $self = shift;
+    return undef if $self->{ terminate };
+    my @next;
+    while ($self->seeing) {
+	my $current = $self->current;
+	print "current = $current\n" if DEBUG;
+	@next = $self->{ graph }->successors( $current );
+	print "next.0 - @next\n" if DEBUG;
+	my %next; @next{ @next } = @next;
+#	delete $next{ $current };
+	print "next.1 - @next\n" if DEBUG;
+	@next = keys %next;
+	my @all = @next;
+	print "all = @all\n" if DEBUG;
+	delete @next{ $self->seen };
+	@next = keys %next;
+	print "next.2 - @next\n" if DEBUG;
+	if (@next) {
+	    @next = $self->{ next_successor }->( $self, \%next );
+	    print "next.3 - @next\n" if DEBUG;
+	    for my $v (@next) {
+		$self->{ tree }->add_edge( $current, $v );
+	    }
+	    if (exists $self->{ pre_edge }) {
+		my $p = $self->{ pre_edge };
+		my $u = $self->current;
+		for my $v (@next) {
+		    $p->( $u, $v, $self, $self->{ state });
+		}
+	    }
+	    last;
+	} else {
+	    $self->visit_postorder;
+	}
+	return undef if $self->{ terminate };
+	$self->_callbacks($current, @all);
+#	delete $next{ $current };
+    }
+    print "next.4 - @next\n" if DEBUG;
+    unless (@next) {
+	unless ( @{ $self->{ roots } } ) {
+	    my $first = $self->{ first_root };
+	    if (defined $first) {
+		@next =
+		    ref $first eq 'CODE' ? 
+			$self->{ first_root }->( $self, $self->{ unseen } ) :
+			$first;
+		return unless @next;
+	    }
+	}
+	unless (@next) {
+	    return unless defined $self->{ next_root };
+	    return unless @next =
+		$self->{ next_root }->( $self, $self->{ unseen } );
+	}
+	return if exists $self->{ seen }->{ $next[0] }; # Sanity check.
+	print "next.5 - @next\n" if DEBUG;
+	push @{ $self->{ roots } }, $next[0];
+    }
+    print "next.6 - @next\n" if DEBUG;
+    if (@next) {
+	$self->visit_preorder( @next );
+    }
+    return $next[0];
+}
+
+sub _order {
+    my ($self, $order) = @_;
+    1 while defined $self->next;
+    my $wantarray = wantarray;
+    if ($wantarray) {
+	@{ $self->{ $order } };
+    } elsif (defined $wantarray) {
+	shift @{ $self->{ $order } };
+    }
+}
+
+sub preorder {
+    my $self = shift;
+    $self->_order( 'preorder' );
+}
+
+sub postorder {
+    my $self = shift;
+    $self->_order( 'postorder' );
+}
+
+sub unseen {
+    my $self = shift;
+    values %{ $self->{ unseen } };
+}
+
+sub seen {
+    my $self = shift;
+    values %{ $self->{ seen } };
+}
+
+sub seeing {
+    my $self = shift;
+    @{ $self->{ order } };
+}
+
+sub roots {
+    my $self = shift;
+    @{ $self->{ roots } };
+}
+
+sub is_root {
+    my ($self, $v) = @_;
+    for my $u (@{ $self->{ roots } }) {
+	return 1 if $u eq $v;
+    }
+    return 0;
+}
+
+sub tree {
+    my $self = shift;
+    $self->{ tree };
+}
+
+sub graph {
+    my $self = shift;
+    $self->{ graph };
+}
+
+sub vertex_by_postorder {
+    my ($self, $i) = @_;
+    exists $self->{ postorder } && $self->{ postorder }->[ $i ];
+}
+
+sub postorder_by_vertex {
+    my ($self, $v) = @_;
+    exists $self->{ postordern } && $self->{ postordern }->{ $v };
+}
+
+sub postorder_vertices {
+    my ($self, $v) = @_;
+    exists $self->{ postordern } ? %{ $self->{ postordern } } : ();
+}
+
+sub vertex_by_preorder {
+    my ($self, $i) = @_;
+    exists $self->{ preorder } && $self->{ preorder }->[ $i ];
+}
+
+sub preorder_by_vertex {
+    my ($self, $v) = @_;
+    exists $self->{ preordern } && $self->{ preordern }->{ $v };
+}
+
+sub preorder_vertices {
+    my ($self, $v) = @_;
+    exists $self->{ preordern } ? %{ $self->{ preordern } } : ();
+}
+
+sub has_state {
+    my ($self, $var) = @_;
+    exists $self->{ state } && exists $self->{ state }->{ $var };
+}
+
+sub get_state {
+    my ($self, $var) = @_;
+    exists $self->{ state } ? $self->{ state }->{ $var } : undef;
+}
+
+sub set_state {
+    my ($self, $var, $val) = @_;
+    $self->{ state }->{ $var } = $val;
+    return 1;
+}
+
+sub delete_state {
+    my ($self, $var) = @_;
+    delete $self->{ state }->{ $var };
+    delete $self->{ state } unless keys %{ $self->{ state } };
+    return 1;
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::Traversal - traverse graphs
+
+=head1 SYNOPSIS
+
+Don't use Graph::Traversal directly, use Graph::Traversal::DFS
+or Graph::Traversal::BFS instead.
+
+    use Graph;
+    my $g = Graph->new;
+    $g->add_edge(...);
+    use Graph::Traversal::...;
+    my $t = Graph::Traversal::...->new(%opt);
+    $t->...
+
+=head1 DESCRIPTION
+
+You can control how the graph is traversed by the various callback
+parameters in the C<%opt>.  In the parameters descriptions below the
+$u and $v are vertices, and the $self is the traversal object itself.
+
+=head2 Callback parameters
+
+The following callback parameters are available:
+
+=over 4
+
+=item tree_edge
+
+Called when traversing an edge that belongs to the traversal tree.
+Called with arguments ($u, $v, $self).
+
+=item non_tree_edge
+
+Called when an edge is met which either leads back to the traversal tree
+(either a C<back_edge>, a C<down_edge>, or a C<cross_edge>).
+Called with arguments ($u, $v, $self).
+
+=item pre_edge
+
+Called for edges in preorder.
+Called with arguments ($u, $v, $self).
+
+=item post_edge
+
+Called for edges in postorder.
+Called with arguments ($u, $v, $self).
+
+=item back_edge
+
+Called for back edges.
+Called with arguments ($u, $v, $self).
+
+=item down_edge
+
+Called for down edges.
+Called with arguments ($u, $v, $self).
+
+=item cross_edge
+
+Called for cross edges.
+Called with arguments ($u, $v, $self).
+
+=item pre
+
+=item pre_vertex
+
+Called for vertices in preorder.
+Called with arguments ($v, $self).
+
+=item post
+
+=item post_vertex
+
+Called for vertices in postorder.
+Called with arguments ($v, $self).
+
+=item first_root
+
+Called when choosing the first root (start) vertex for traversal.
+Called with arguments ($self, $unseen) where $unseen is a hash
+reference with the unseen vertices as keys.
+
+=item next_root
+
+Called when choosing the next root (after the first one) vertex for
+traversal (useful when the graph is not connected).  Called with
+arguments ($self, $unseen) where $unseen is a hash reference with
+the unseen vertices as keys.  If you want only the first reachable
+subgraph to be processed, set the next_root to C<undef>.
+
+=item start
+
+Identical to defining C<first_root> and undefining C<next_root>.
+
+=item next_alphabetic
+
+Set this to true if you want the vertices to be processed in
+alphabetic order (and leave first_root/next_root undefined).
+
+=item next_numeric
+
+Set this to true if you want the vertices to be processed in
+numeric order (and leave first_root/next_root undefined).
+
+=item next_successor
+
+Called when choosing the next vertex to visit.  Called with arguments
+($self, $next) where $next is a hash reference with the possible
+next vertices as keys.  Use this to provide a custom ordering for
+choosing vertices, as opposed to C<next_numeric> or C<next_alphabetic>.
+
+=back
+
+The parameters C<first_root> and C<next_successor> have a 'hierarchy'
+of how they are determined: if they have been explicitly defined, use
+that value.  If not, use the value of C<next_alphabetic>, if that has
+been defined.  If not, use the value of C<next_numeric>, if that has
+been defined.  If not, the next vertex to be visited is chose randomly.
+
+=head2 Methods
+
+The following methods are available:
+
+=over 4
+
+=item unseen
+
+Return the unseen vertices in random order.
+
+=item seen
+
+Return the seen vertices in random order.
+
+=item seeing
+
+Return the active fringe vertices in random order.
+
+=item preorder
+
+Return the vertices in preorder traversal order.
+
+=item postorder
+
+Return the vertices in postorder traversal order.
+
+=item vertex_by_preorder
+
+    $v = $t->vertex_by_preorder($i)
+
+Return the ith (0..$V-1) vertex by preorder.
+
+=item preorder_by_vertex
+
+    $i = $t->preorder_by_vertex($v)
+
+Return the preorder index (0..$V-1) by vertex.
+
+=item vertex_by_postorder
+
+    $v = $t->vertex_by_postorder($i)
+
+Return the ith (0..$V-1) vertex by postorder.
+
+=item postorder_by_vertex
+
+    $i = $t->postorder_by_vertex($v)
+
+Return the postorder index (0..$V-1) by vertex.
+
+=item preorder_vertices
+
+Return a hash with the vertices as the keys and their preorder indices
+as the values.
+
+=item postorder_vertices
+
+Return a hash with the vertices as the keys and their postorder
+indices as the values.
+
+=item tree
+
+Return the traversal tree as a graph.
+
+=item has_state
+
+    $t->has_state('s')
+
+Test whether the traversal has state 's' attached to it.
+
+=item get_state
+
+    $t->get_state('s')
+
+Get the state 's' attached to the traversal (C<undef> if none).
+
+=item set_state
+
+    $t->set_state('s', $s)
+
+Set the state 's' attached to the traversal.
+
+=item delete_state
+
+    $t->delete_state('s')
+
+Delete the state 's' from the traversal.
+
+=back
+
+=head2 Backward compatibility
+
+The following parameters are for backward compatibility to Graph 0.2xx:
+
+=over 4
+
+=item get_next_root
+
+Like C<next_root>.
+
+=item successor
+
+Identical to having C<tree_edge> both C<non_tree_edge> defined
+to be the same.
+
+=item unseen_successor
+
+Like C<tree_edge>.
+
+=item seen_successor
+
+Like C<seed_edge>.
+
+=back
+
+=head2 Special callbacks
+
+If in a callback you call the special C<terminate> method,
+the traversal is terminated, no more vertices are traversed.
+
+=head1 SEE ALSO
+
+L<Graph::Traversal::DFS>, L<Graph::Traversal::BFS>
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
diff --git a/perllib/Graph/Traversal/BFS.pm b/perllib/Graph/Traversal/BFS.pm
new file mode 100644
index 00000000..2678f72e
--- /dev/null
+++ b/perllib/Graph/Traversal/BFS.pm
@@ -0,0 +1,59 @@
+package Graph::Traversal::BFS;
+
+use strict;
+
+use Graph::Traversal;
+use base 'Graph::Traversal';
+
+sub current {
+    my $self = shift;
+    $self->{ order }->[ 0 ];
+}
+
+sub see {
+    my $self = shift;
+    shift @{ $self->{ order } };
+}
+
+*bfs = \&Graph::Traversal::postorder;
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::Traversal::BFS - breadth-first traversal of graphs
+
+=head1 SYNOPSIS
+
+    use Graph;
+    my $g = Graph->new;
+    $g->add_edge(...);
+    use Graph::Traversal::BFS;
+    my $b = Graph::Traversal::BFS->new(%opt);
+    $b->bfs; # Do the traversal.
+
+=head1 DESCRIPTION
+
+With this class one can traverse a Graph in breadth-first order.
+
+The callback parameters %opt are explained in L<Graph::Traversal>.
+
+=head2 Methods
+
+The following methods are available:
+
+=over 4
+
+=item dfs
+
+Traverse the graph in depth-first order.
+
+=back
+
+=head1 SEE ALSO
+
+L<Graph::Traversal>, L<Graph::Traversal::DFS>, L<Graph>.
+
+=cut
diff --git a/perllib/Graph/Traversal/DFS.pm b/perllib/Graph/Traversal/DFS.pm
new file mode 100644
index 00000000..4b109bd8
--- /dev/null
+++ b/perllib/Graph/Traversal/DFS.pm
@@ -0,0 +1,59 @@
+package Graph::Traversal::DFS;
+
+use strict;
+
+use Graph::Traversal;
+use base 'Graph::Traversal';
+
+sub current {
+    my $self = shift;
+    $self->{ order }->[ -1 ];
+}
+
+sub see {
+    my $self = shift;
+    pop @{ $self->{ order } };
+}
+
+*dfs = \&Graph::Traversal::postorder;
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::Traversal::DFS - depth-first traversal of graphs
+
+=head1 SYNOPSIS
+
+    use Graph;
+    my $g = Graph->new;
+    $g->add_edge(...);
+    use Graph::Traversal::DFS;
+    my $d = Graph::Traversal::DFS->new(%opt);
+    $d->dfs; # Do the traversal.
+
+=head1 DESCRIPTION
+
+With this class one can traverse a Graph in depth-first order.
+
+The callback parameters %opt are explained in L<Graph::Traversal>.
+
+=head2 Methods
+
+The following methods are available:
+
+=over 4
+
+=item dfs
+
+Traverse the graph in depth-first order.
+
+=back
+
+=head1 SEE ALSO
+
+L<Graph::Traversal>, L<Graph::Traversal::BFS>, L<Graph>.
+
+=cut
diff --git a/perllib/Graph/Undirected.pm b/perllib/Graph/Undirected.pm
new file mode 100644
index 00000000..3993bb1c
--- /dev/null
+++ b/perllib/Graph/Undirected.pm
@@ -0,0 +1,49 @@
+package Graph::Undirected;
+
+use Graph;
+use base 'Graph';
+use strict;
+
+=pod
+
+=head1 NAME
+
+Graph::Undirected - undirected graphs
+
+=head1 SYNOPSIS
+
+    use Graph::Undirected;
+    my $g = Graph::Undirected->new;
+
+    # Or alternatively:
+
+    use Graph;
+    my $g = Graph->new(undirected => 1);
+    my $g = Graph->new(directed => 0);
+
+=head1 DESCRIPTION
+
+Graph::Undirected allows you to create undirected graphs.
+
+For the available methods, see L<Graph>.
+
+=head1 SEE ALSO
+
+L<Graph>, L<Graph::Directed>
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
+
+sub new {
+    my $class = shift;
+    bless Graph->new(undirected => 1, @_), ref $class || $class;
+}
+
+1;
diff --git a/perllib/Graph/UnionFind.pm b/perllib/Graph/UnionFind.pm
new file mode 100644
index 00000000..83a921f0
--- /dev/null
+++ b/perllib/Graph/UnionFind.pm
@@ -0,0 +1,183 @@
+package Graph::UnionFind;
+
+use strict;
+
+sub _PARENT  () { 0 }
+sub _RANK    () { 1 }
+
+sub new {
+    my $class = shift;
+    bless { }, $class;
+}
+
+sub add {
+    my ($self, $elem) = @_;
+    $self->{ $elem } = [ $elem, 0 ];
+}
+
+sub has {
+    my ($self, $elem) = @_;
+    exists $self->{ $elem };
+}
+
+sub _parent {
+    return undef unless defined $_[1];
+    if (@_ == 2) {
+	exists $_[0]->{ $_[ 1 ] } ? $_[0]->{ $_[1] }->[ _PARENT ] : undef;
+    } elsif (@_ == 3) {
+	$_[0]->{ $_[1] }->[ _PARENT ] = $_[2];
+    } else {
+	require Carp;
+	Carp::croak(__PACKAGE__ . "::_parent: bad arity");
+    }
+}
+
+sub _rank {
+    return unless defined $_[1];
+    if (@_ == 2) {
+	exists $_[0]->{ $_[1] } ? $_[0]->{ $_[1] }->[ _RANK ] : undef;
+    } elsif (@_ == 3) {
+	$_[0]->{ $_[1] }->[ _RANK ] = $_[2];
+    } else {
+	require Carp;
+	Carp::croak(__PACKAGE__ . "::_rank: bad arity");
+    }
+}
+
+sub find {
+    my ($self, $x) = @_;
+    my $px = $self->_parent( $x );
+    return unless defined $px;
+    $self->_parent( $x, $self->find( $px ) ) if $px ne $x;
+    $self->_parent( $x );
+}
+
+sub union {
+    my ($self, $x, $y) = @_;
+    $self->add($x) unless $self->has($x);
+    $self->add($y) unless $self->has($y);
+    my $px = $self->find( $x );
+    my $py = $self->find( $y );
+    return if $px eq $py;
+    my $rx = $self->_rank( $px );
+    my $ry = $self->_rank( $py );
+    # print "union($x, $y): px = $px, py = $py, rx = $rx, ry = $ry\n";
+    if ( $rx > $ry ) {
+	$self->_parent( $py, $px );
+    } else {
+	$self->_parent( $px, $py );
+	$self->_rank( $py, $ry + 1 ) if $rx == $ry;
+    }
+}
+
+sub same {
+    my ($uf, $u, $v) = @_;
+    my $fu = $uf->find($u);
+    return undef unless defined $fu;
+    my $fv = $uf->find($v);
+    return undef unless defined $fv;
+    $fu eq $fv;
+}
+
+1;
+__END__
+=pod
+
+=head1 NAME
+
+Graph::UnionFind - union-find data structures
+
+=head1 SYNOPSIS
+
+    use Graph::UnionFind;
+    my $uf = Graph::UnionFind->new;
+
+    # Add the vertices to the data structure.
+    $uf->add($u);
+    $uf->add($v);
+
+    # Join the partitions of the vertices.
+    $uf->union( $u, $v );
+
+    # Find the partitions the vertices belong to
+    # in the union-find data structure.  If they
+    # are equal, they are in the same partition.
+    # If the vertex has not been seen,
+    # undef is returned.
+    my $pu = $uf->find( $u );
+    my $pv = $uf->find( $v );
+    $uf->same($u, $v) # Equal to $pu eq $pv. 
+
+    # Has the union-find seen this vertex?
+    $uf->has( $v )
+
+=head1 DESCRIPTION
+
+I<Union-find> is a special data structure that can be used to track the
+partitioning of a set into subsets (a problem known also as I<disjoint sets>).
+
+Graph::UnionFind() is used for Graph::connected_components(),
+Graph::connected_component(), and Graph::same_connected_components()
+if you specify a true C<union_find> parameter when you create an undirected
+graph.
+
+Note that union-find is one way: you cannot (easily) 'ununion'
+vertices once you have 'unioned' them.  This means that if you
+delete edges from a C<union_find> graph, you will get wrong results
+from the Graph::connected_components(), Graph::connected_component(),
+and Graph::same_connected_components().
+
+=head2 API
+
+=over 4
+
+=item add
+
+    $uf->add($v)
+
+Add the vertex v to the union-find.
+
+=item union
+
+    $uf->union($u, $v)
+
+Add the edge u-v to the union-find.  Also implicitly adds the vertices.
+
+=item has
+
+    $uf->has($v)
+
+Return true if the vertex v has been added to the union-find, false otherwise.
+
+=item find
+
+    $uf->find($v)
+
+Return the union-find partition the vertex v belongs to,
+or C<undef> if it has not been added.
+
+=item new
+
+    $uf = Graph::UnionFind->new()
+
+The constructor.
+
+=item same
+
+    $uf->same($u, $v)
+
+Return true of the vertices belong to the same union-find partition
+the vertex v belongs to, false otherwise.
+
+=back
+
+=head1 AUTHOR AND COPYRIGHT
+
+Jarkko Hietaniemi F<jhi@iki.fi>
+
+=head1 LICENSE
+
+This module is licensed under the same terms as Perl itself.
+
+=cut
+
diff --git a/perllib/Heap071/Elem.pm b/perllib/Heap071/Elem.pm
new file mode 100644
index 00000000..40ae5dc9
--- /dev/null
+++ b/perllib/Heap071/Elem.pm
@@ -0,0 +1,159 @@
+package Heap071::Elem;
+
+use strict;
+use vars qw($VERSION @ISA @EXPORT @EXPORT_OK);
+
+require Exporter;
+require AutoLoader;
+
+@ISA = qw(Exporter AutoLoader);
+
+# No names exported.
+# No names available for export.
+
+@EXPORT = ( );
+
+$VERSION = '0.71';
+
+
+# Preloaded methods go here.
+
+# new will usually be superceded by child,
+# but provide an empty hash as default and
+# accept any provided filling for it.
+sub new {
+    my $self = shift;
+    my $class = ref($self) || $self;
+
+    return bless { heap=>undef, @_ }, $class;
+}
+
+sub heap {
+    my $self = shift;
+    @_ ? ($self->{heap} = shift) : $self->{heap};
+}
+
+sub cmp {
+    die "This cmp method must be superceded by one that knows how to compare elements."
+}
+
+# Autoload methods go after =cut, and are processed by the autosplit program.
+
+1;
+__END__
+# Below is the stub of documentation for your module. You better edit it!
+
+=head1 NAME
+
+Heap::Elem - Perl extension for elements to be put in Heaps
+
+=head1 SYNOPSIS
+
+  use Heap::Elem::SomeInheritor;
+
+  use Heap::SomeHeapClass;
+
+  $elem = Heap::Elem::SomeInheritor->new( $value );
+  $heap = Heap::SomeHeapClass->new;
+
+  $heap->add($elem);
+
+=head1 DESCRIPTION
+
+This is an inheritable class for Heap Elements.  It provides
+the interface documentation and some inheritable methods.
+Only a child classes can be used - this class is not complete.
+
+=head1 METHODS
+
+=over 4
+
+=item $elem = Heap::Elem::SomeInheritor->new( [args] );
+
+Creates a new Elem.
+
+=item $elem->heap( $val ); $elem->heap;
+
+Provides a method for use by the Heap processing routines.
+If a value argument is provided, it will be saved.  The
+new saved value is always returned.  If no value argument
+is provided, the old saved value is returned.
+
+The Heap processing routines use this method to map an element
+into its internal structure.  This is needed to support the
+Heap methods that affect elements that are not are the top
+of the heap - I<decrease_key> and I<delete>.
+
+The Heap processing routines will ensure that this value is
+undef when this elem is removed from a heap, and is not undef
+after it is inserted into a heap.  This means that you can
+check whether an element is currently contained within a heap
+or not.  (It cannot be used to determine which heap an element
+is contained in, if you have multiple heaps.  Keeping that
+information accurate would make the operation of merging two
+heaps into a single one take longer - it would have to traverse
+all of the elements in the merged heap to update them; for
+Binomial and Fibonacci heaps that would turn an O(1) operation
+into an O(n) one.)
+
+=item $elem1->cmp($elem2)
+
+A routine to compare two elements.  It must return a negative
+value if this element should go higher on the heap than I<$elem2>,
+0 if they are equal, or a positive value if this element should
+go lower on the heap than I<$elem2>.  Just as with sort, the
+Perl operators <=> and cmp cause the smaller value to be returned
+first; similarly you can negate the meaning to reverse the order
+- causing the heap to always return the largest element instead
+of the smallest.
+
+=back
+
+=head1 INHERITING
+
+This class can be inherited to provide an oject with the
+ability to be heaped.  If the object is implemented as
+a hash, and if it can deal with a key of I<heap>, leaving
+it unchanged for use by the heap routines, then the following
+implemetation will work.
+
+  package myObject;
+
+  require Exporter;
+
+  @ISA = qw(Heap::Elem);
+
+  sub new {
+      my $self = shift;
+      my $class = ref($self) || $self;
+
+      my $self = SUPER::new($class);
+
+      # set $self->{key} = $value;
+  }
+
+  sub cmp {
+      my $self = shift;
+      my $other = shift;
+
+      $self->{key} cmp $other->{key};
+  }
+
+  # other methods for the rest of myObject's functionality
+
+=head1 AUTHOR
+
+John Macdonald, jmm@perlwolf.com
+
+=head1 COPYRIGHT
+
+Copyright 1998-2003, O'Reilly & Associates.
+
+This code is distributed under the same copyright terms as perl itself.
+
+=head1 SEE ALSO
+
+Heap(3), Heap::Elem::Num(3), Heap::Elem::NumRev(3),
+Heap::Elem::Str(3), Heap::Elem::StrRev(3).
+
+=cut
diff --git a/perllib/Heap071/Fibonacci.pm b/perllib/Heap071/Fibonacci.pm
new file mode 100644
index 00000000..3308bf31
--- /dev/null
+++ b/perllib/Heap071/Fibonacci.pm
@@ -0,0 +1,482 @@
+package Heap071::Fibonacci;
+
+use strict;
+use vars qw($VERSION @ISA @EXPORT @EXPORT_OK);
+
+require Exporter;
+require AutoLoader;
+
+@ISA = qw(Exporter AutoLoader);
+
+# No names exported.
+# No names available for export.
+@EXPORT = ( );
+
+$VERSION = '0.71';
+
+
+# Preloaded methods go here.
+
+# common names
+#	h	- heap head
+#	el	- linkable element, contains user-provided value
+#	v	- user-provided value
+
+################################################# debugging control
+
+my $debug = 0;
+my $validate = 0;
+
+# enable/disable debugging output
+sub debug {
+    @_ ? ($debug = shift) : $debug;
+}
+
+# enable/disable validation checks on values
+sub validate {
+    @_ ? ($validate = shift) : $validate;
+}
+
+my $width = 3;
+my $bar = ' | ';
+my $corner = ' +-';
+my $vfmt = "%3d";
+
+sub set_width {
+    $width = shift;
+    $width = 2 if $width < 2;
+
+    $vfmt = "%${width}d";
+    $bar = $corner = ' ' x $width;
+    substr($bar,-2,1) = '|';
+    substr($corner,-2,2) = '+-';
+}
+
+sub hdump;
+
+sub hdump {
+    my $el = shift;
+    my $l1 = shift;
+    my $b = shift;
+
+    my $ch;
+    my $ch1;
+
+    unless( $el ) {
+	print $l1, "\n";
+	return;
+    }
+
+    hdump $ch1 = $el->{child},
+	$l1 . sprintf( $vfmt, $el->{val}->val),
+	$b . $bar;
+
+    if( $ch1 ) {
+	for( $ch = $ch1->{right}; $ch != $ch1; $ch = $ch->{right} ) {
+	    hdump $ch, $b . $corner, $b . $bar;
+	}
+    }
+}
+
+sub heapdump {
+    my $h;
+
+    while( $h = shift ) {
+	my $top = $$h or last;
+	my $el = $top;
+
+	do {
+	    hdump $el, sprintf( "%02d: ", $el->{degree}), '    ';
+	    $el = $el->{right};
+	} until $el == $top;
+	print "\n";
+    }
+}
+
+sub bhcheck;
+
+sub bhcheck {
+    my $el = shift;
+    my $p = shift;
+
+    my $cur = $el;
+    my $prev;
+    my $ch;
+    do {
+	$prev = $cur;
+	$cur = $cur->{right};
+	die "bad back link" unless $cur->{left} == $prev;
+	die "bad parent link"
+	    unless (defined $p && defined $cur->{p} && $cur->{p} == $p)
+		|| (!defined $p && !defined $cur->{p});
+	die "bad degree( $cur->{degree} > $p->{degree} )"
+	    if $p && $p->{degree} <= $cur->{degree};
+	die "not heap ordered"
+	    if $p && $p->{val}->cmp($cur->{val}) > 0;
+	$ch = $cur->{child} and bhcheck $ch, $cur;
+    } until $cur == $el;
+}
+
+
+sub heapcheck {
+    my $h;
+    my $el;
+    while( $h = shift ) {
+	heapdump $h if $validate >= 2;
+	$el = $$h and bhcheck $el, undef;
+    }
+}
+
+
+################################################# forward declarations
+
+sub ascending_cut;
+sub elem;
+sub elem_DESTROY;
+sub link_to_left_of;
+
+################################################# heap methods
+
+# Cormen et al. use two values for the heap, a pointer to an element in the
+# list at the top, and a count of the number of elements.  The count is only
+# used to determine the size of array required to hold log(count) pointers,
+# but perl can set array sizes as needed and doesn't need to know their size
+# when they are created, so we're not maintaining that field.
+sub new {
+    my $self = shift;
+    my $class = ref($self) || $self;
+    my $h = undef;
+    bless \$h, $class;
+}
+
+sub DESTROY {
+    my $h = shift;
+
+    elem_DESTROY $$h;
+}
+
+sub add {
+    my $h = shift;
+    my $v = shift;
+    $validate && do {
+	die "Method 'heap' required for element on heap"
+	    unless $v->can('heap');
+	die "Method 'cmp' required for element on heap"
+	    unless $v->can('cmp');
+    };
+    my $el = elem $v;
+    my $top;
+    if( !($top = $$h) ) {
+	$$h = $el;
+    } else {
+	link_to_left_of $top->{left}, $el ;
+	link_to_left_of $el,$top;
+	$$h = $el if $v->cmp($top->{val}) < 0;
+    }
+}
+
+sub top {
+    my $h = shift;
+    $$h && $$h->{val};
+}
+
+*minimum = \&top;
+
+sub extract_top {
+    my $h = shift;
+    my $el = $$h or return undef;
+    my $ltop = $el->{left};
+    my $cur;
+    my $next;
+
+    # $el is the heap with the lowest value on it
+    # move all of $el's children (if any) to the top list (between
+    # $ltop and $el)
+    if( $cur = $el->{child} ) {
+	# remember the beginning of the list of children
+	my $first = $cur;
+	do {
+	    # the children are moving to the top, clear the p
+	    # pointer for all of them
+	    $cur->{p} = undef;
+	} until ($cur = $cur->{right}) == $first;
+
+	# remember the end of the list
+	$cur = $cur->{left};
+	link_to_left_of $ltop, $first;
+	link_to_left_of $cur, $el;
+    }
+
+    if( $el->{right} == $el ) {
+	# $el had no siblings or children, the top only contains $el
+	# and $el is being removed
+	$$h = undef;
+    } else {
+	link_to_left_of $el->{left}, $$h = $el->{right};
+	# now all those loose ends have to be merged together as we
+	# search for the
+	# new smallest element
+	$h->consolidate;
+    }
+
+    # extract the actual value and return that, $el is no longer used
+    # but break all of its links so that it won't be pointed to...
+    my $top = $el->{val};
+    $top->heap(undef);
+    $el->{left} = $el->{right} = $el->{p} = $el->{child} = $el->{val} =
+	undef;
+    $top;
+}
+
+*extract_minimum = \&extract_top;
+
+sub absorb {
+    my $h = shift;
+    my $h2 = shift;
+
+    my $el = $$h;
+    unless( $el ) {
+	$$h = $$h2;
+	$$h2 = undef;
+	return $h;
+    }
+
+    my $el2 = $$h2 or return $h;
+
+    # add $el2 and its siblings to the head list for $h
+    # at start, $ell -> $el -> ... -> $ell is on $h (where $ell is
+    #				$el->{left})
+    #           $el2l -> $el2 -> ... -> $el2l are on $h2
+    # at end, $ell -> $el2l -> ... -> $el2 -> $el -> ... -> $ell are
+    #				all on $h
+    my $el2l = $el2->{left};
+    link_to_left_of $el->{left}, $el2;
+    link_to_left_of $el2l, $el;
+
+    # change the top link if needed
+    $$h = $el2 if $el->{val}->cmp( $el2->{val} ) > 0;
+
+    # clean out $h2
+    $$h2 = undef;
+
+    # return the heap
+    $h;
+}
+
+# a key has been decreased, it may have to percolate up in its heap
+sub decrease_key {
+    my $h = shift;
+    my $top = $$h;
+    my $v = shift;
+    my $el = $v->heap or return undef;
+    my $p;
+
+    # first, link $h to $el if it is now the smallest (we will
+    # soon link $el to $top to properly put it up to the top list,
+    # if it isn't already there)
+    $$h = $el if $top->{val}->cmp( $v ) > 0;
+
+    if( $p = $el->{p} and $v->cmp($p->{val}) < 0 ) {
+	# remove $el from its parent's list - it is now smaller
+
+	ascending_cut $top, $p, $el;
+    }
+
+    $v;
+}
+
+
+# to delete an item, we bubble it to the top of its heap (as if its key
+# had been decreased to -infinity), and then remove it (as in extract_top)
+sub delete {
+    my $h = shift;
+    my $v = shift;
+    my $el = $v->heap or return undef;
+
+    # if there is a parent, cut $el to the top (as if it had just had its
+    # key decreased to a smaller value than $p's value
+    my $p;
+    $p = $el->{p} and ascending_cut $$h, $p, $el;
+
+    # $el is in the top list now, make it look like the smallest and
+    # remove it
+    $$h = $el;
+    $h->extract_top;
+}
+
+
+################################################# internal utility functions
+
+sub elem {
+    my $v = shift;
+    my $el = undef;
+    $el = {
+	p	=>	undef,
+	degree	=>	0,
+	mark	=>	0,
+	child	=>	undef,
+	val	=>	$v,
+	left	=>	undef,
+	right	=>	undef,
+    };
+    $el->{left} = $el->{right} = $el;
+    $v->heap($el);
+    $el;
+}
+
+sub elem_DESTROY {
+    my $el = shift;
+    my $ch;
+    my $next;
+    $el->{left}->{right} = undef;
+
+    while( $el ) {
+	$ch = $el->{child} and elem_DESTROY $ch;
+	$next = $el->{right};
+
+	defined $el->{val} and $el->{val}->heap(undef);
+	$el->{child} = $el->{right} = $el->{left} = $el->{p} = $el->{val}
+	    = undef;
+	$el = $next;
+    }
+}
+
+sub link_to_left_of {
+    my $l = shift;
+    my $r = shift;
+
+    $l->{right} = $r;
+    $r->{left} = $l;
+}
+
+sub link_as_parent_of {
+    my $p = shift;
+    my $c = shift;
+
+    my $pc;
+
+    if( $pc = $p->{child} ) {
+	link_to_left_of $pc->{left}, $c;
+	link_to_left_of $c, $pc;
+    } else {
+	link_to_left_of $c, $c;
+    }
+    $p->{child} = $c;
+    $c->{p} = $p;
+    $p->{degree}++;
+    $c->{mark} = 0;
+    $p;
+}
+
+sub consolidate {
+    my $h = shift;
+
+    my $cur;
+    my $this;
+    my $next = $$h;
+    my $last = $next->{left};
+    my @a;
+    do {
+	# examine next item on top list
+	$this = $cur = $next;
+	$next = $cur->{right};
+	my $d = $cur->{degree};
+	my $alt;
+	while( $alt = $a[$d] ) {
+	    # we already saw another item of the same degree,
+	    # put the larger valued one under the smaller valued
+	    # one - switch $cur and $alt if necessary so that $cur
+	    # is the smaller
+	    ($cur,$alt) = ($alt,$cur)
+		if $cur->{val}->cmp( $alt->{val} ) > 0;
+	    # remove $alt from the top list
+	    link_to_left_of $alt->{left}, $alt->{right};
+	    # and put it under $cur
+	    link_as_parent_of $cur, $alt;
+	    # make sure that $h still points to a node at the top
+	    $$h = $cur;
+	    # we've removed the old $d degree entry
+	    $a[$d] = undef;
+	    # and we now have a $d+1 degree entry to try to insert
+	    # into @a
+	    ++$d;
+	}
+	# found a previously unused degree
+	$a[$d] = $cur;
+    } until $this == $last;
+    $cur = $$h;
+    for $cur (grep defined, @a) {
+	$$h = $cur if $$h->{val}->cmp( $cur->{val} ) > 0;
+    }
+}
+
+sub ascending_cut {
+    my $top = shift;
+    my $p = shift;
+    my $el = shift;
+
+    while( 1 ) {
+	if( --$p->{degree} ) {
+	    # there are still other children below $p
+	    my $l = $el->{left};
+	    $p->{child} = $l;
+	    link_to_left_of $l, $el->{right};
+	} else {
+	    # $el was the only child of $p
+	    $p->{child} = undef;
+	}
+	link_to_left_of $top->{left}, $el;
+	link_to_left_of $el, $top;
+	$el->{p} = undef;
+	$el->{mark} = 0;
+
+	# propagate up the list
+	$el = $p;
+
+	# quit at the top
+	last unless $p = $el->{p};
+
+	# quit if we can mark $el
+	$el->{mark} = 1, last unless $el->{mark};
+    }
+}
+
+
+1;
+
+__END__
+
+=head1 NAME
+
+Heap::Fibonacci - a Perl extension for keeping data partially sorted
+
+=head1 SYNOPSIS
+
+  use Heap::Fibonacci;
+
+  $heap = Heap::Fibonacci->new;
+  # see Heap(3) for usage
+
+=head1 DESCRIPTION
+
+Keeps elements in heap order using a linked list of Fibonacci trees.
+The I<heap> method of an element is used to store a reference to
+the node in the list that refers to the element.
+
+See L<Heap> for details on using this module.
+
+=head1 AUTHOR
+
+John Macdonald, jmm@perlwolf.com
+
+=head1 COPYRIGHT
+
+Copyright 1998-2003, O'Reilly & Associates.
+
+This code is distributed under the same copyright terms as perl itself.
+
+=head1 SEE ALSO
+
+Heap(3), Heap::Elem(3).
+
+=cut