add method IsDigraphAutomorphism

[ChristopherRussell]

This operation takes a digraph and a permutation and returns true if the
permutation is an automorphism of the digraph. See also #105.

[wilfwilson]

Hi Chris, it's a nice idea to have this function.

This currently only works for digraphs without multiple edges. It returns the wrong answer sometimes for multidigraphs. The plan is to remove multidigraphs at some point in the future, but for now we should support them I think.

For example:

<multidigraph with 3 vertices, 3 edges>
gap> IsDigraphAutomorphism(gr, (2, 3));
true

Vertices 2 and 3 are not equivalent, since they have different numbers of out-neighbours. So (2, 3) should not be an automorphism. Also, somewhat related, (2, 3) is not in the automorphism group of the multidigraph, i.e. if you do (2, 3) in AutomorphismGroup(gr); then the answer is false.

Remember that automorphism groups for digraphs and multidigraphs are different (see the doc of AutomorphismGroup).

For non-multidigraphs, I reckon the best method would be to do return OnDigraphs(gr, p) = gr; rather than what you're doing now, which involves constructing the list of edges and repeated searching in the list of edges.

For multidigraphs, I'm not sure. Perhaps require the automorphism to consist of two perms (one of the vertices, one of the edges?) and then run OnMultiDigraphs rather than OnDigraphs? Something for you to think about.

I also think the documentation for IsDigraphAutomorphism could maybe explain what an automorphism is (or at least provide a reference toAutomorphismGroup if you think that's sufficient).

[wilfwilson]

Please first add a check for whether gr is a multidigraph, if so, then have an error message saying it doesn't work for digraphs with multiple edges.

[wilfwilson]

Please change it to:

...of <C>gr</C> for all vertices u and v in digraph. (To quantify the vertices)