License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.STACS.2020.43
URN: urn:nbn:de:0030-drops-119046
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Fuhlbrück, Frank ; Köbler, Johannes ; Verbitsky, Oleg

Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

LIPIcs-STACS-2020-43.pdf (0.6 MB)


It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges.
Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.

BibTeX - Entry

  author =	{Frank Fuhlbr{\"u}ck and Johannes K{\"o}bler and Oleg Verbitsky},
  title =	{{Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-119046},
  doi =		{10.4230/LIPIcs.STACS.2020.43},
  annote =	{Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-F{\"u}rer-Immerman Graphs, coherent Configurations}

Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-Fürer-Immerman Graphs, coherent Configurations
Collection: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue Date: 2020
Date of publication: 04.03.2020

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