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DOI: 10.4230/LIPIcs.ICALP.2020.73

The Iteration Number of Colour Refinement

Authors: Sandra Kiefer and Brendan D. McKay

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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.

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Sandra Kiefer and Brendan D. McKay. The Iteration Number of Colour Refinement. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 73:1-73:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kiefer_et_al:LIPIcs.ICALP.2020.73,
  author =	{Kiefer, Sandra and McKay, Brendan D.},
  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 =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.73},
  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}
}

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The Iteration Number of Colour Refinement

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