When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.45
URN: urn:nbn:de:0030-drops-109893
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10989/
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### The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs

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### Abstract

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.

### BibTeX - Entry

```@InProceedings{kiefer_et_al:LIPIcs:2019:10989,
author =	{Sandra Kiefer and Daniel Neuen},
title =	{{The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs}},
booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages =	{45:1--45:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-117-7},
ISSN =	{1868-8969},
year =	{2019},
volume =	{138},
editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},