License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.134
URN: urn:nbn:de:0030-drops-142035
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14203/
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Grohe, Martin ; Kiefer, Sandra

Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

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LIPIcs-ICALP-2021-134.pdf (0.8 MB)

Abstract

The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm.
We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs.
The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth.

BibTeX - Entry

@InProceedings{grohe_et_al:LIPIcs.ICALP.2021.134,
  author =	{Grohe, Martin and Kiefer, Sandra},
  title =	{{Logarithmic Weisfeiler-Leman Identifies All Planar Graphs}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{134:1--134:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14203},
  URN =		{urn:nbn:de:0030-drops-142035},
  doi =		{10.4230/LIPIcs.ICALP.2021.134},
  annote =	{Keywords: Weisfeiler-Leman algorithm, finite-variable logic, isomorphism testing, planar graphs, quantifier depth, iteration number}
}

Keywords: Weisfeiler-Leman algorithm, finite-variable logic, isomorphism testing, planar graphs, quantifier depth, iteration number
Collection: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Issue Date: 2021
Date of publication: 02.07.2021

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