The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs
Sandra Kiefer, Daniel Neuen ยท SIAM Journal on Discrete Mathematics ยท 2022
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 two-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 nonisomorphic graphs is at most the dimension required to distinguish nonisomorphic 3-connected components of the graphs (assuming dimension at least 2). To obtain our decomposition result, we show that, for ๐ โฅ 2, the ๐-dimensional algorithm distinguishes ๐-separators, i.e., ๐-tuples of vertices that separate the graph, from other vertex ๐-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 ๐ on the Weisfeiler--Leman dimension of the class of graphs of treewidth at most ๐. Using a construction by Cai, Fรผrer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2. (A corrected version is attached.)