Does there exist a Graph Isomorphism Algorithm that uses Local Isomorphism to construct a Global Isomorphism?
For example, two graphs are given, say, $H, G$. it is asked to determine whether $G\simeq H$.
Now assume there exists an algorithm that splits graph $G$ in to Subgraph $G_1, G_2$(in $G$, $G_1$ and $G_2$ are connected through a set of edges, say, $R_1$) and finds Isomorphism to corresponding graph $H_1 , H_2$(Local Isomorphism), then merge those Isomorphism to check whether $G\simeq H$ (Global isomorphism).
Does there exist such algorithm in Graph Isomorphism Literature ?
I apologize for a vague description, I should be more specific, but it is really hard to specify at the moment. For the time being, note that I do not mean a splitting based on vertex classification entirely or something like that. An algorithm that is based on $K$ Dimensional Weisfeiler-lehman(depth $K$ neighborhoods) entirely, is not what I meant, that is for sure**. if $K=1$, then it can not handle regular graph. You can "construct" graph that makes $K$ Dimensional Weisfeiler-lehman ineffective, for example Cai-Furer-Immerman pair. Certainly I do not mean that.Note that NAUTY uses 1 Dimensional Weisfeiler-Lehman method.
This question is the closest I got to mine, but it does not answer my query.