research-article

The Weisfeiler--Leman Dimension of Planar Graphs Is at Most 3

Authors:

Ilia Ponomarenko,

Pascal SchweitzerAuthors Info & Claims

Journal of the ACM (JACM), Volume 66, Issue 6

Article No.: 44, Pages 1 - 31

https://doi.org/10.1145/3333003

Published: 27 November 2019 Publication History

Abstract

We prove that the Weisfeiler--Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best-known upper bounds for the dimension and number of variables were 14 and 15, respectively.

First, we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of every arc-colored 3-connected graph belonging to this class.

Then, we prove that, apart from several exceptional graphs (which have WL-dimension at most 2), the individualization of two appropriately chosen vertices of a colored 3-connected planar graph followed by the one-dimensional WL-algorithm produces the discrete vertex partition. This implies that the three-dimensional WL-algorithm determines the orbits of arc-colored 3-connected planar graphs.

As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3.

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Cited By

Kiefer SNeuen DSobocinski PLago UEsparza J(2024)Bounding the Weisfeiler-Leman Dimension via a Depth Analysis of I/R-TreesProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662122(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662122
Neuen D(2024)Isomorphism Testing for Graphs Excluding Small Topological SubgraphsACM Transactions on Algorithms10.1145/365198620:3(1-43)Online publication date: 21-Jun-2024
https://dl.acm.org/doi/10.1145/3651986
Focke JGoldberg LRoth MZivný S(2024)Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-ComplexityProceedings of the ACM on Management of Data10.1145/36516142:2(1-17)Online publication date: 14-May-2024
https://dl.acm.org/doi/10.1145/3651614
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Index Terms

The Weisfeiler--Leman Dimension of Planar Graphs Is at Most 3
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
  2. Logic
    1. Finite Model Theory

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 66, Issue 6

December 2019

231 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3368192

Editor:
Éva Tardos
Cornell University

Issue’s Table of Contents

Copyright © 2019 ACM.

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Association for Computing Machinery

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Publication History

Published: 27 November 2019

Accepted: 01 May 2019

Revised: 01 April 2019

Received: 01 February 2018

Published in JACM Volume 66, Issue 6

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Cited By

Kiefer SNeuen DSobocinski PLago UEsparza J(2024)Bounding the Weisfeiler-Leman Dimension via a Depth Analysis of I/R-TreesProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662122(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662122
Neuen D(2024)Isomorphism Testing for Graphs Excluding Small Topological SubgraphsACM Transactions on Algorithms10.1145/365198620:3(1-43)Online publication date: 21-Jun-2024
https://dl.acm.org/doi/10.1145/3651986
Focke JGoldberg LRoth MZivný S(2024)Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-ComplexityProceedings of the ACM on Management of Data10.1145/36516142:2(1-17)Online publication date: 14-May-2024
https://dl.acm.org/doi/10.1145/3651614
Collins NLevet M(2024)Count-free Weisfeiler–Leman and group isomorphismInternational Journal of Algebra and Computation10.1142/S021819672450010334:03(283-330)Online publication date: 3-Apr-2024
https://doi.org/10.1142/S0218196724500103
Guo JGavrilyuk APonomarenko I(2024)On the Weisfeiler–Leman Dimension of Permutation GraphsSIAM Journal on Discrete Mathematics10.1137/23M157501938:2(1915-1929)Online publication date: 21-Jun-2024
https://doi.org/10.1137/23M1575019
Verbitsky OZhukovskii M(2024)Canonization of a Random Circulant Graph by Counting WalksWALCOM: Algorithms and Computation10.1007/978-981-97-0566-5_23(319-334)Online publication date: 18-Mar-2024
https://dl.acm.org/doi/10.1007/978-981-97-0566-5_23
Böker JLevie RHuang NVillar SMorris COh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Fine-grained expressivity of graph neural networksProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668144(46658-46700)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668144
Dimitrov RZhao ZAbboud RCeylan İOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)PLANEProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666827(16028-16054)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3666827
Morris CGeerts FTönshoff JGrohe MKrause ABrunskill ECho KEngelhardt BSabato SScarlett J(2023)WL meet VCProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3619459(25275-25302)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.5555/3618408.3619459
Grochow JLevet M(2023)On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)Electronic Proceedings in Theoretical Computer Science10.4204/EPTCS.390.12390(185-202)Online publication date: 30-Sep-2023
https://doi.org/10.4204/EPTCS.390.12
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