bugs - IsomorphicGraphQ fails to recognize that two empty graphs are identical

Bug introduced in 9.0.1 or earlier and fixed in 11.0.1

Consider the following two graphs:

g1 = Graph[{1, 2, 3}, {}];

g2 = SimpleGraph@Graph[{1, 2, 3}, {1 <-> 1}];

They are clearly the same thing because SimpleGraph will remove the self loop (only edge) from g2.

Yet IsomorphicGraphQ disagrees:

IsomorphicGraphQ[g1, g2]

(* False *)

I believe this is a bug and it exists in all of 9.0.1 through to 10.4.0.

Is there a workaround?

Answer

A simple workaround is to re-build the graph object by cycling it through some other representation. Here are two possible solutions:

rebuildGraph[g_] := Uncompress@Compress[g] (* solution 1 *)
rebuildGraph[g_] := Graph[VertexList[g], EdgeList[g]] (* solution 2 destroys properties but it's fine for isomorphism testing purposes *)

isomorphicGraphQ[g1_, g2_] := 
 IsomorphicGraphQ[rebuildGraph[g1], rebuildGraph[g2]]

Now this gives a correct result:

isomorphicGraphQ[g1, g2]
(* True *)

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