Find Mapping Node in a Graph
You want to solve the problem of graph isomorphism (GI). GI is not known to be in P or NP-complete; that is, we do not know any efficient algorithms.

Many algorithms have been proposed for GI. None applies to all graphs and is always fast, but you may be able to find one that is sufficiently fast in your application.

In 2015, Babai proposed a quasi-polynomial algorithm for GI [1]. As far as I know, peer-reviewed is not yet completed; I also do not know if it is practically feasible. If correct, it is the most "efficient" algorithm known in terms of Landau-style complexity.




Graph Isomorphism in Quasipolynomial Time by L. Babai (2015) [retraction note]