research-article

Testing first-order properties for subclasses of sparse graphs

Authors:

Zdeněk Dvořák,

Robin ThomasAuthors Info & Claims

Journal of the ACM (JACM), Volume 60, Issue 5

Article No.: 36, Pages 1 - 24

https://doi.org/10.1145/2499483

Published: 28 October 2013 Publication History

Abstract

We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion.

More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings.

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Cited By

Gajarský JPilipczuk MSokołowski MStamoulis GToruńczyk SSobocinski PLago UEsparza J(2024)Elementary first-order model checking for sparse graphsProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662094(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662094
Bonnet ÉGiocanti Ude Mendez PSimon PThomassé SToruńczyk S(2024)Twin-Width IV: Ordered Graphs and MatricesJournal of the ACM10.1145/365115171:3(1-45)Online publication date: 11-Mar-2024
https://dl.acm.org/doi/10.1145/3651151
Dreier JMählmann NToruńczyk SMohar BShinkar IO'Donnell R(2024)Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph ClassesProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649739(1550-1560)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649739
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Index Terms

Testing first-order properties for subclasses of sparse graphs
1. Theory of computation
  1. Design and analysis of algorithms

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cover image Journal of the ACM

Journal of the ACM Volume 60, Issue 5

October 2013

245 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2528384

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Copyright © 2013 ACM.

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Association for Computing Machinery

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Publication History

Published: 28 October 2013

Accepted: 01 June 2013

Revised: 01 January 2013

Received: 01 September 2011

Published in JACM Volume 60, Issue 5

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Seventh Framework Programme
project 1M0545 of Ministry of Education of Czech Republic (ZD)
Division of Mathematical Sciences

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Cited By

Gajarský JPilipczuk MSokołowski MStamoulis GToruńczyk SSobocinski PLago UEsparza J(2024)Elementary first-order model checking for sparse graphsProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662094(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662094
Bonnet ÉGiocanti Ude Mendez PSimon PThomassé SToruńczyk S(2024)Twin-Width IV: Ordered Graphs and MatricesJournal of the ACM10.1145/365115171:3(1-45)Online publication date: 11-Mar-2024
https://dl.acm.org/doi/10.1145/3651151
Dreier JMählmann NToruńczyk SMohar BShinkar IO'Donnell R(2024)Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph ClassesProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649739(1550-1560)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649739
Grobler MJiang YOssona de Mendez PSiebertz SVigny A(2024)Discrepancy and sparsityJournal of Combinatorial Theory, Series B10.1016/j.jctb.2024.06.001169(96-133)Online publication date: Nov-2024
https://doi.org/10.1016/j.jctb.2024.06.001
Dvořák ZKrál' DThomas R(2024)Three-coloring triangle-free graphs on surfaces VI. 3-colorability of quadrangulationsJournal of Combinatorial Theory, Series B10.1016/j.jctb.2023.11.001164(517-548)Online publication date: Jan-2024
https://doi.org/10.1016/j.jctb.2023.11.001
Fomin FGolovach PStamoulis GThilikos D(2023)An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOLACM Transactions on Computation Theory10.1145/357127814:3-4(1-29)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1145/3571278
Toruńczyk S(2023)Flip-width: Cops and Robber on dense graphs2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00045(663-700)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00045
Jiang YNešetřil JOssona de Mendez P(2023)From χ- to χ-bounded classesJournal of Combinatorial Theory, Series B10.1016/j.jctb.2021.05.006158(186-209)Online publication date: Jan-2023
https://doi.org/10.1016/j.jctb.2021.05.006
Nešetřil JOssona de Mendez PSiebertz S(2023)Modulo-counting first-order logic on bounded expansion classesDiscrete Mathematics10.1016/j.disc.2023.113700(113700)Online publication date: Sep-2023
https://doi.org/10.1016/j.disc.2023.113700
Dreier JGajarský JKiefer SPilipczuk MToruńczyk S(2022)Treelike Decompositions for Transductions of Sparse GraphsProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3533349(1-14)Online publication date: 2-Aug-2022
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