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.ICALP.2020.73
URN: urn:nbn:de:0030-drops-124801
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12480/
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Kiefer, Sandra ; McKay, Brendan D.

The Iteration Number of Colour Refinement

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LIPIcs-ICALP-2020-73.pdf (0.6 MB)

Abstract

The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph.
A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ≥ 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation.

BibTeX - Entry

@InProceedings{kiefer_et_al:LIPIcs:2020:12480,
  author =	{Sandra Kiefer and Brendan D. McKay},
  title =	{{The Iteration Number of Colour Refinement}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{73:1--73:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12480},
  URN =		{urn:nbn:de:0030-drops-124801},
  doi =		{10.4230/LIPIcs.ICALP.2020.73},
  annote =	{Keywords: Colour Refinement, iteration number, Weisfeiler-Leman algorithm, quantifier depth}
}

Keywords: Colour Refinement, iteration number, Weisfeiler-Leman algorithm, quantifier depth
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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