The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

Abstract

The Weisfeiler-Leman procedure is a widely-used technique for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. This implies that the dimension of the algorithm needed to distinguish two given non-isomorphic graphs is at most the dimension required to distinguish non-isomorphic 3-connected components of the graphs (assuming dimension at least 2). To obtain our decomposition result, we show that, for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of the class of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.