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

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 IsomorphicGraphQand 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