Isomorphism on Subgraph-Closed Graph Classes: A Complexity Dichotomy and Intermediate Graph Classes

Abstract

We study the graph isomorphism problem for graph classes defined by sets of forbidden subgraphs. We show that there is a complexity dichotomy in case the set of forbidden subgraphs is finite. More precisely, we show that the problem is polynomial-time solvable if the forbidden set contains a forest of subdivided stars and is graph isomorphism complete otherwise. We also show that, assuming that the graph isomorphism problem is not polynomial-time solvable in general, there is no such dichotomy for the cases of infinite sets of forbidden subgraphs. To this end, we conditionally show that there exists a graph class closed under taking subgraphs with intermediate isomorphism problem, i.e., a class on which the isomorphism problem is neither polynomial-time solvable nor graph isomorphism complete.