Testing Graph Isomorphism in Parallel by Playing a Game

Our starting point is the observation that if graphs in a class C have low descriptive complexity in first order logic, then the isomorphism problem for C is solvable by a fast parallel algorithm (essentially, by a simple combinatorial algorithm known as the multidimensional Weisfeiler-Lehman algorithm). Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC1, answering an open question posed by Chandrasekharan. 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 Weisfeiler-Lehman algorithm is new. Combined with other known results, it also yields a new AC1 isomorphism algorithm for planar graphs.