research-article

Deciding first-order properties of nowhere dense graphs

Authors:

Stephan Kreutzer,

Sebastian SiebertzAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 89 - 98

https://doi.org/10.1145/2591796.2591851

Published: 31 May 2014 Publication History

Abstract

Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [30], form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion.

We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption).

As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those.

Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.

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Cited By

Agrawal ABhattacharjee SJana SSahu A(2023)Parameterized complexity of perfectly matched setsTheoretical Computer Science10.1016/j.tcs.2023.113861958(113861)Online publication date: May-2023
https://doi.org/10.1016/j.tcs.2023.113861
Adler IBenediktsson BMacpherson D(2022)Vapnik–Chervonenkis dimension and density on Johnson and Hamming graphsDiscrete Applied Mathematics10.1016/j.dam.2021.09.017312:C(29-44)Online publication date: 15-May-2022
https://dl.acm.org/doi/10.1016/j.dam.2021.09.017
Toruńczyk SSuciu DTao YWei Z(2020)Aggregate Queries on Sparse DatabasesProceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3375395.3387660(427-443)Online publication date: 14-Jun-2020
https://dl.acm.org/doi/10.1145/3375395.3387660
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Index Terms

Deciding first-order properties of nowhere dense graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Graph enumeration
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

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Published: 31 May 2014

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Cited By

Agrawal ABhattacharjee SJana SSahu A(2023)Parameterized complexity of perfectly matched setsTheoretical Computer Science10.1016/j.tcs.2023.113861958(113861)Online publication date: May-2023
https://doi.org/10.1016/j.tcs.2023.113861
Adler IBenediktsson BMacpherson D(2022)Vapnik–Chervonenkis dimension and density on Johnson and Hamming graphsDiscrete Applied Mathematics10.1016/j.dam.2021.09.017312:C(29-44)Online publication date: 15-May-2022
https://dl.acm.org/doi/10.1016/j.dam.2021.09.017
Toruńczyk SSuciu DTao YWei Z(2020)Aggregate Queries on Sparse DatabasesProceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3375395.3387660(427-443)Online publication date: 14-Jun-2020
https://dl.acm.org/doi/10.1145/3375395.3387660
Kierstead HYang DYi J(2020)On coloring numbers of graph powersDiscrete Mathematics10.1016/j.disc.2019.111712343:6(111712)Online publication date: Jul-2020
https://doi.org/10.1016/j.disc.2019.111712
Dvořák ZCzumaj A(2018)Thin graph classes and polynomial-time approximation schemesProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175415(1685-1701)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175415
Kreutzer SRabinovich RSiebertz S(2018)Polynomial Kernels and Wideness Properties of Nowhere Dense Graph ClassesACM Transactions on Algorithms10.1145/327465215:2(1-19)Online publication date: 16-Nov-2018
https://dl.acm.org/doi/10.1145/3274652
Berkholz CKeppeler JSchweikardt N(2018)Answering FO+MOD Queries under Updates on Bounded Degree DatabasesACM Transactions on Database Systems10.1145/323205643:2(1-32)Online publication date: 22-Aug-2018
https://dl.acm.org/doi/10.1145/3232056
Akhoondian Amiri SOssona de Mendez PRabinovich RSiebertz SScheideler CFineman J(2018)Distributed Domination on Graph Classes of Bounded ExpansionProceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures10.1145/3210377.3210383(143-151)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3210377.3210383
Pilipczuk MSiebertz SToruńczyk S(2018)On the number of types in sparse graphsProceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3209108.3209178(799-808)Online publication date: 9-Jul-2018
https://dl.acm.org/doi/10.1145/3209108.3209178
Schweikardt NSegoufin LVigny AVan den Bussche JArenas M(2018)Enumeration for FO Queries over Nowhere Dense GraphsProceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3196959.3196971(151-163)Online publication date: 27-May-2018
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