License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.45
URN: urn:nbn:de:0030-drops-109893
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10989/
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Kiefer, Sandra ; Neuen, Daniel

The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs

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LIPIcs-MFCS-2019-45.pdf (0.5 MB)

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},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10989},
  URN =		{urn:nbn:de:0030-drops-109893},
  doi =		{10.4230/LIPIcs.MFCS.2019.45},
  annote =	{Keywords: Weisfeiler-Leman, separators, first-order logic, counting quantifiers}
}

Keywords: Weisfeiler-Leman, separators, first-order logic, counting quantifiers
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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