I am trying to enumerate all non-isomorphic graphs of size n and found this question: Enumerate all non-isomorphic graphs of a certain size
The accepted answer describes a method to do this:
Assume we know all non-isomorphic graphs of size n-1. Take each of them and add a new vertex in all possible ways. Then for all these graphs calculate a canonical labeling and check whether the new vertex has label 1 (or is in the same orbit of the automorphism group as the vertex with label 1). In that case, save the graph. Otherwise stop and try the next extension of the graph with n-1 vertices.
However, there's a problem: Say n=3 and we are currently considering the graph P2:
Now there are 4 ways to add a third vertex:
However, two of them create the same graph, P3. Using a canonical labeling as explained above doesn't solve this issue. So the extensions themselves need to be canonical, somehow. I assume I need to use the automorphism group of the graph of size n-1 somehow.
So that's my question. How can I make sure that all my extensions give non-isomorphic graphs?
I know how to find canonical labelings and automorphism group generators, so your answer can use them as much as you like :)
Thanks!
