Testing Graph Isomorphism in Parallel by Playing a Game

Abstract

Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in ${\rm TC^1}\subseteq\mbox{\rm NC\ensuremath{^{2}}}$. If no counting quantifiers are needed, then Graph Isomorphism for C is even in AC¹. The definability conditions can be checked by designing a winning strategy for suitable Ehrenfeucht-Fraïssé games with a logarithmic number of rounds. The parallel isomorphism algorithm this approach yields is a simple combinatorial algorithm known as the Weisfeiler-Lehman (WL) algorithm.

Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC¹, answering an open question from [9]. Furthermore, we obtain an AC¹ algorithm for testing isomorphism of rotation systems (combinatorial specifications of graph embeddings). The AC¹ upper bound was known before, but the fact that this bound can be achieved by the simple WL algorithm is new. Combined with other known results, it also yields a new AC¹ isomorphism algorithm for planar graphs.