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.2019.117
URN: urn:nbn:de:0030-drops-106931
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10693/
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Grohe, Martin ; Kiefer, Sandra

A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus

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Abstract

The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers.
It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor [M. Grohe, 2017]. However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 [S. Kiefer et al., 2017].
In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus g is at most 4g+3. For the WL dimension of graphs embeddable in an orientable surface of Euler genus g, our approach yields an upper bound of 2g + 3.

BibTeX - Entry

@InProceedings{grohe_et_al:LIPIcs:2019:10693,
  author =	{Martin Grohe and Sandra Kiefer},
  title =	{{A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{117:1--117:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10693},
  URN =		{urn:nbn:de:0030-drops-106931},
  doi =		{10.4230/LIPIcs.ICALP.2019.117},
  annote =	{Keywords: Weisfeiler-Leman algorithm, finite-variable logic, isomorphism testing, planar graphs, bounded genus}
}

Keywords: Weisfeiler-Leman algorithm, finite-variable logic, isomorphism testing, planar graphs, bounded genus
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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