Structural and logical approaches to the graph isomorphism problem

The question of whether there is a polynomial time algorithm deciding whether two graphs are isomorphic has been a one of the best known open problems in theoretical computer science for more than forty years. Indeed, the graph isomorphism problem is one of the very few natural problems in NP that is neither known to be in P nor known to be NP-complete. The question is still wide open, but a number of deep partial results giving polynomial time algorithms for specific classes of graphs are known. Many of them have been obtained through a group theoretic approach that dominated the research on the graph isomorphism problem since the early 1980s. After an introductory survey on the graph isomorphism problem, in my talk I will focus on approaches based on structural graph theory and connections between logical definability and certain combinatorial algorithms for the isomorphism problem. In particular, I will speak about two recent results: The first says that the Weisfeiler-Lehman algorithm (a simple combinatorial algorithm) can be used to decide isomorphism on graph classes with excluded minors in polynomial time. The second says that isomorphism can be decided in polynomial time on graph classes with excluded topological subgraphs. ∗Humboldt-Universität zu Berlin. 188 Copyright © SIAM. Unauthorized reproduction of this article is prohibited.