# combinatorial enumerative: generate all non-isomorphic graphs connected n n vertices module local complementation?

I would like to generate a list of all the simple, connected and non-directed graphics of $$n$$ vertices, standard module of the isomorphism graph, and local complementation module., which is the next operation: for a graph $$G = (V, E)$$mark a vertex $$v in V$$, and let $$S_v$$ be your neighborhood (such that $$v not in S_v$$). Leave $$C_v$$ Be a complete graph about the vertices. $$S_v$$. Then define $$G & # 39; = G mathrm {xor} C_v$$, that is, for any pair of vertices in the vicinity of $$v$$, we deny the existence of an edge.

An example of this operation can be seen in fig. 1 in arXiv: 0710.2243.

by local edge complementationIt seems that there is a way to list all these graphs efficiently, see the sequence of numbers at https://oeis.org/A156800.

I know the program `geng`, what gives me all the non-isomorphic graphs of $$n$$ vertices. I currently use it with Mathematica along with a custom comparator that, given a graphic $$G_1$$, check if it is isomorphic to any of all possible local add-ons $$G_2$$; unfortunately that is slow due to the extensive use of `IsomorphicGraphQ`and because there are a lot of local add-ons for any given graphic.

Does anyone know of a more efficient way to generate such graphics? Maybe directly using something inside `nauty`?

Thank you!
– JB