Graph isomorphism, general remarks

Abstract

An open question is the computational complexity of recognizing when two graphs are isomorphic. In an attempt to answer this question we shall analyze the relative computational complexity of generalizations and restrictions of the graph isomorphism problem. We show graph isomorphism of regular undirected graphs is complete over isomorphism of explicitly given structures (say Tarski models from logic). We also show a fundamental difference between how automorphism groups can act on a graph of valence n and how they can act on graphs of valence n + 1 (with one exception). This group theoretic result seems to have implications on the role of valence for graph isomorphism algorithms. Finally, we introduce “certificates” for symmetric cubic graphs.