Towards an Isomorphism Dichotomy for Hereditary Graph Classes

TL;DR

This paper advances the understanding of the graph isomorphism problem by establishing a dichotomy for hereditary graph classes defined by two forbidden subgraphs, introducing new techniques for structural analysis and reductions.

Contribution

It develops a unifying framework for reductions, generalizes modular decomposition to colored graphs, and extends algorithms for graphs with bounded generalized color valence.

Findings

Resolved the complexity classification for most hereditary graph classes with two forbidden subgraphs.

Introduced a new methodology for proving isomorphism completeness via a general reduction framework.

Extended polynomial-time algorithms to graphs with bounded generalized color valence.

Abstract

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.