Graph Isomorphism for Graph Classes Characterized by two Forbidden Induced Subgraphs

TL;DR

This paper investigates the computational complexity of the Graph Isomorphism problem within graph classes defined by two forbidden induced subgraphs, establishing a near-complete classification into polynomial-time solvable and isomorphism-complete cases.

Contribution

It provides a dichotomy theorem for the complexity of Graph Isomorphism on classes characterized by two forbidden subgraphs, with new structural analysis techniques and reductions for remaining open cases.

Findings

Dichotomy into polynomial-time solvable and isomorphism-complete classes for most cases.

Equivalence of forbidding a pan and forbidding a triangle in terms of Graph Isomorphism complexity.

Structural techniques for analyzing graph classes with forbidden induced subgraphs.

Abstract

We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further reducing the remaining open cases we show that (with respect to graph isomorphism) forbidding a pan is equivalent to forbidding a clique of size three.