Fix output of is_chordal for empty graphs (#6563)

* Fix for is_chordal for empty graphs

* Handle self loops case

2 files changed, 10 insertions, 4 deletions

diff --git a/networkx/algorithms/chordal.py b/networkx/algorithms/chordal.py
index 6ff8b046..444bfbc6 100644
--- a/networkx/algorithms/chordal.py
+++ b/networkx/algorithms/chordal.py
@@ -73,6 +73,8 @@ def is_chordal(G):
     search. It returns False when it finds that the separator for any node
     is not a clique.  Based on the algorithms in [1]_.
 
+    Self loops are ignored.
+
     References
     ----------
     .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms
@@ -80,6 +82,8 @@ def is_chordal(G):
        selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984),
        pp. 566–579.
     """
+    if len(G.nodes) <= 3:
+        return True
     return len(_find_chordality_breaker(G)) == 0
 
 
@@ -130,6 +134,8 @@ def find_induced_nodes(G, s, t, treewidth_bound=sys.maxsize):
     The algorithm is inspired by Algorithm 4 in [1]_.
     A formal definition of induced node can also be found on that reference.
 
+    Self Loops are ignored
+
     References
     ----------
     .. [1] Learning Bounded Treewidth Bayesian Networks.
@@ -326,9 +332,9 @@ def _find_chordality_breaker(G, s=None, treewidth_bound=sys.maxsize):
 
     If it does find one, it returns (u,v,w) where u,v,w are the three
     nodes that together with s are involved in the cycle.
+
+    It ignores any self loops.
     """
-    if nx.number_of_selfloops(G) > 0:
-        raise nx.NetworkXError("Input graph is not chordal.")
     unnumbered = set(G)
     if s is None:
         s = arbitrary_element(G)
diff --git a/networkx/algorithms/tests/test_chordal.py b/networkx/algorithms/tests/test_chordal.py
index 132698a9..148b22f2 100644
--- a/networkx/algorithms/tests/test_chordal.py
+++ b/networkx/algorithms/tests/test_chordal.py
@@ -60,11 +60,11 @@ class TestMCS:
         assert not nx.is_chordal(self.non_chordal_G)
         assert nx.is_chordal(self.chordal_G)
         assert nx.is_chordal(self.connected_chordal_G)
+        assert nx.is_chordal(nx.Graph())
         assert nx.is_chordal(nx.complete_graph(3))
         assert nx.is_chordal(nx.cycle_graph(3))
         assert not nx.is_chordal(nx.cycle_graph(5))
-        with pytest.raises(nx.NetworkXError, match="Input graph is not chordal"):
-            nx.is_chordal(self.self_loop_G)
+        assert nx.is_chordal(self.self_loop_G)
 
     def test_induced_nodes(self):
         G = nx.generators.classic.path_graph(10)