`Digraph(Graph(G))` may not return a graph isomorphic to `G`

[bojankuzma000]

When I construct a complete bipartite digraph $K_{1,5}$ and then turn it into a Grape package Graph, and then back into a Digraphs package Digraph, I would expect to get an isomorphic digraph. But apparently not so.

gap> G:=CompleteBipartiteDigraph(1,5); 
<immutable complete bipartite digraph with bicomponent sizes 1 and 5>
gap> IsomorphismDigraphs(Digraph(Graph(G)),G);
fail

Even the adjacency matrix is different. The following

gap> Display(AdjacencyMatrix(G));
[ [  0,  1,  1,  1,  1,  1 ],
  [  1,  0,  0,  0,  0,  0 ],
  [  1,  0,  0,  0,  0,  0 ],
  [  1,  0,  0,  0,  0,  0 ],
  [  1,  0,  0,  0,  0,  0 ],
  [  1,  0,  0,  0,  0,  0 ] ]

is as expected for the bipartite digraph G. But:

gap> Display(AdjacencyMatrix(Digraph(Graph(G))));
[ [  0,  1,  1,  1,  1,  1 ],
  [  1,  0,  1,  1,  1,  1 ],
  [  1,  1,  0,  1,  1,  1 ],
  [  1,  1,  1,  0,  1,  1 ],
  [  1,  1,  1,  1,  0,  1 ],
  [  1,  0,  0,  0,  0,  0 ] ]

is a different adjacency matrix. Why?

Bojan

[bojankuzma000]

In contrast: CompleteBipartiteDigraph(4,1) works as expected:

gap> G:=CompleteBipartiteDigraph(4,1);
<immutable complete bipartite digraph with bicomponent sizes 4 and 1>
gap> IsomorphismDigraphs(Digraph(Graph(G)),G);
()

Also, $K_{2,4}$ works as expected:

gap> G:=CompleteBipartiteDigraph(2,4);
<immutable complete bipartite digraph with bicomponent sizes 2 and 4>
gap> IsomorphismDigraphs(Digraph(Graph(G)),G);
()

[james-d-mitchell]

@bojankuzma000 thanks for the report, we will investigate and get back to you!

[wilfwilson]

Oh dear, thank you for your report @bojankuzma000. This seems to be the underlying cause, I'll quickly make a fix and a new release:

gap> G:=CompleteBipartiteDigraph(1,5);
<immutable complete bipartite digraph with bicomponent sizes 1 and 5>
gap> DigraphBicomponents(G);
[ [ 1 ], [ 2, 3, 4, 5, 6 ] ]
gap> AutomorphismGroup(G);
Group([ (1,2,3,4,5), (1,2) ])

The automorphism group of a complete bipartite digraph is set at construction, and should be the direct product of the symmetric groups on the bicomponents (perhaps additionally being able to swap the bicomponents), so we are obviously just doing that wrong!

[wilfwilson]

Simply put, our code was assuming that for DirectProduct(SymmetricGroup([1]), SymmetricGroup([2..6])), GAP would return SymmetricGroup([2..6]). But in fact, GAP returns (the isomorphic, but not identical) SymmetricGroup([1..5]).