The Power of the WeisfeilerLeman Algorithm to Decompose Graphs
Abstract
The WeisfeilerLeman procedure is a widelyused approach for graph isomorphism testing that works by iteratively computing an isomorphisminvariant 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 3connected components. We prove that the 2dimensional WeisfeilerLeman algorithm implicitly computes the decomposition of a graph into its 3connected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3connected components (assuming it is at least 2). This result implies that for k >= 2, the kdimensional algorithm distinguishes kseparators, i.e., ktuples of vertices that separate the graph, from other vertex ktuples. 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 WeisfeilerLeman dimension of graphs of treewidth at most k. Using a construction by Cai, Fürer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.05268
 Bibcode:
 2019arXiv190805268K
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Logic in Computer Science;
 Mathematics  Combinatorics
 EPrint:
 30 pages, 4 figures, full version of a paper accepted at MFCS 2019