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.

References

[1]

Vikraman Arvind, Johannes Köbler, Gaurav Rattan, and Oleg Verbitsky. 2015. On the power of color refinement. In Proceedings of the Future Technologies Conference (FCT’15), Lecture Notes in Computer Science, Vol. 9210. Springer, 339--350.

[2]

Albert Atserias and Elitza N. Maneva. 2013. Sherali-Adams relaxations and indistinguishability in counting logics. SIAM J. Comput. 42, 1 (2013), 112--137.

Digital Library

[3]

László Babai, Anuj Dawar, Pascal Schweitzer, and Jacobo Torán. 2015. The graph isomorphism problem (Dagstuhl Seminar 15511). Dagstuhl Rep. 5, 12 (2015), 1--17.

[4]

Christoph Berkholz and Jakob Nordström. 2016. Near-optimal lower bounds on quantifier depth and Weisfeiler-Leman refinement steps. In Proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS’16). ACM, 267--276.

Digital Library

[5]

Jin-yi Cai, Martin Fürer, and Neil Immerman. 1992. An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 4 (1992), 389--410.

[6]

Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner. 2009. Planar graph isomorphism is in log-space. In Proceedings of the IEEE Conference on Computational Complexity. IEEE Computer Society, 203--214.

Digital Library

[7]

Sergei Evdokimov and Ilia N. Ponomarenko. 2000. Separability number and schurity number of coherent configurations. Electr. J. Combin. 7 (2000).

[8]

Martin Grohe. 1998. Fixed-point logics on planar graphs. In Proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS’98). IEEE Computer Society, 6--15.

[9]

Martin Grohe. 2000. Isomorphism testing for embeddable graphs through definability. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC’00). ACM, 63--72.

Digital Library

[10]

Martin Grohe. 2012. Fixed-point definability and polynomial time on graphs with excluded minors. J. ACM 59, 5 (2012), 27:1--27:64.

Digital Library

[11]

Martin Grohe. 2016. Tangled up in blue (a survey on connectivity, decompositions, and tangles). CoRR abs/1605.06704 (2016).

[12]

Martin Grohe. 2017. Descriptive Complexity, Canonisation, and Definable Graph Structure Theory. Cambridge University Press.

[13]

Martin Grohe and Julian Mariño. 1999. Definability and descriptive complexity on databases of bounded tree-width. In Proceedings of the International Conference on Database Theory (ICDT’99), Lecture Notes in Computer Science, Vol. 1540. Springer, 70--82.

[14]

Martin Grohe and Martin Otto. 2012. Pebble games and linear equations. In Proceedings of the International Workshop/Annual Conference of the Computer Science Logic (CSL’12), LIPIcs, Vol. 16. Schloss Dagstuhl--Leibniz-Zentrum für Informatik, 289--304.

[15]

Martin Grohe and Oleg Verbitsky. 2006. Testing graph isomorphism in parallel by playing a game. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’06) (Lecture Notes in Computer Science), Vol. 4051. Springer, 3--14.

Digital Library

[16]

Mahdieh Hasheminezhad, Brendan D. McKay, and Tristan Reeves. 2011. Recursive generation of simple planar 5-regular graphs and pentangulations. J. Graph Algor. Appl. 15, 3 (2011), 417--436.

[17]

John E. Hopcroft and Robert Endre Tarjan. 1971. A V² algorithm for determining isomorphism of planar graphs. Inf. Process. Lett. 1, 1 (1971), 32--34.

[18]

John E. Hopcroft and Robert Endre Tarjan. 1972. Isomorphism of planar graphs. In Complexity of Computer Computations (The IBM Research Symposia Series). Plenum Press, 131--152.

[19]

John E. Hopcroft and Robert Endre Tarjan. 1973. A V log V algorithm for isomorphism of triconnected planar graphs. J. Comput. System Sci. 7, 3 (1973), 323--331.

Digital Library

[20]

John E. Hopcroft and J. K. Wong. 1974. Linear time algorithm for isomorphism of planar graphs (preliminary report). In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC’74). ACM, 172--184.

[21]

Dmitri Karpov and Alexey Pastor. 2003. On the structure of a k-connected graph. J. Math. Sci. 113, 4 (2003), 584--597.

[22]

Sandra Kiefer and Pascal Schweitzer. 2016. Upper bounds on the quantifier depth for graph differentiation in first order logic. In Proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS’16). ACM, 287--296.

Digital Library

[23]

Sandra Kiefer, Pascal Schweitzer, and Erkal Selman. 2015. Graphs identified by logics with counting. In Proceedings of the International Symposium on Mathematical Foundations of Computer Science (MFCS’15), Lecture Notes in Computer Science, Vol. 9234. Springer, 319--330.

[24]

Mikhail Klin, Christian Pech, and Sven Reichard. 2018. COCO2P: GAP-package for the Computation with Coherent Configurations, Version 0.17. Retrieved from https://github.com/chpech/COCO2P.

[25]

Stephen G. Kobourov. 2013. Force-directed drawing algorithms. In Handbook of Graph Drawing and Visualization. Chapman 8 Hall/CRC, 383--408.

[26]

Brendan D. McKay and Adolfo Piperno. 2014. Practical graph isomorphism, II. J. Symb. Comput. 60 (2014), 94--112.

Digital Library

[27]

Jenish C. Mehta. 2014. Dynamic complexity of planar 3-connected graph isomorphism. In Proceedings of the International Computer Science Symposium in Russia (CSR’14), Lecture Notes in Computer Science, Vol. 8476. Springer, 273--286.

[28]

Bojan Mohar and Carsten Thomassen. 2001. Graphs on Surfaces. Johns Hopkins University Press, Baltimore, MD.

[29]

Ilia N. Ponomarenko. 1991. The isomorphism problem for classes of graphs closed under contraction. J. Math. Sci. 55, 2 (1991), 1621--1643.

[30]

Joachim Redies. 2014. Defining PTIME Problems on Planar Graphs with Few Variables. Master’s thesis. RWTH Aachen University.

[31]

William Thomas Tutte. 1963. How to draw a graph. Proc. Lond. Math. Soc. (3) 13 (1963), 743--767.

[32]

Oleg Verbitsky. 2007. Planar graphs: Logical complexity and parallel isomorphism tests. In STACS’07, Lecture Notes in Computer Science, Vol. 4393. Springer, 682--693.

[33]

Boris Weisfeiler. 1976. On Construction and Identification of Graphs. Lecture Notes in Mathematics, Vol. 558. Springer.

Cited By

Torán JWörz F(2024)Cutting Planes Width and the Complexity of Graph Isomorphism RefutationsACM Transactions on Computational Logic10.1145/367712125:4(1-25)Online publication date: 18-Jul-2024
https://dl.acm.org/doi/10.1145/3677121
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
Show More Cited By

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.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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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

Torán JWörz F(2024)Cutting Planes Width and the Complexity of Graph Isomorphism RefutationsACM Transactions on Computational Logic10.1145/367712125:4(1-25)Online publication date: 18-Jul-2024
https://dl.acm.org/doi/10.1145/3677121
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
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