eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-08-20
45:1
45:15
10.4230/LIPIcs.MFCS.2019.45
article
The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
Kiefer, Sandra
1
https://orcid.org/0000-0003-4614-9444
Neuen, Daniel
1
https://orcid.org/0000-0002-4940-0318
RWTH Aachen University, Aachen, Germany
The Weisfeiler-Leman procedure is a widely-used approach 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 bi- and triconnected components.
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its triconnected 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 3-connected components (assuming dimension at least 2).
This result implies 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 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol138-mfcs2019/LIPIcs.MFCS.2019.45/LIPIcs.MFCS.2019.45.pdf
Weisfeiler-Leman
separators
first-order logic
counting quantifiers