Twin-width I: tractable FO model checking

Bonnet, Édouard; Kim, Eun Jung; Thomassé, Stéphan; Watrigant, Rémi

doi:10.48550/arxiv.2004.14789

Cited by 7 publications

(44 citation statements)

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“…Our main result also resolves a question of Bonnet, Geniet, Kim, Thomassé, and Watrigant [7] concerning sparse twin-width; see [7][8][9] for the definition and background on (sparse) twin-width. Bonnet et al [7] proved that graphs with bounded stack-number have bounded sparse twin-width, and they write that they "believe that the inclusion is strict"; that is, there exists a class of graphs with bounded sparse twin-width and unbounded stack-number.…”

Section: Reflectionssupporting

confidence: 75%

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Stack-number is not bounded by queue-number

Dujmović,

Eppstein,

Hickingbotham

et al. 2020

Preprint

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Section: Reflectionssupporting

confidence: 75%

“…Apply this result with G := (S b ) b∈N and H := (H n ) n∈N . Every star graph has twin-width 0 and planar graphs have bounded twin-width[9]. Since S b H n is 7-degenerate, it contains no K 8,8 subgraph.…”

mentioning

confidence: 99%

Stack-number is not bounded by queue-number

Dujmović,

Eppstein,

Hickingbotham

et al. 2020

Preprint

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show abstract

“…This, combined with the result of [BKTW20b] yields fixed-parameter tractability of the modelchecking problem for classes of ordered graphs of bounded twin-width, proving the implication (1)→(5). We also prove the converse implication, under the common complexity-theoretic assumption FPT =AW[ * ].…”

Section: Resultsmentioning

confidence: 69%

Ordered graphs of bounded twin-width

Simon,

Toruńczyk

2021

Preprint

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We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove that those are exactly the classes which avoid certain large grid-like structures and induced substructures. From this we derive that the model-checking problem for first-order logic is fixed-parameter tractable over a hereditary class of ordered graphs if, and -under common complexity-theoretic assumptions -only if the class has bounded twin-width. We also show that bounded twin-width is equivalent to the NIP property from model theory, as well as the smallness condition from enumerative combinatorics. We prove the existence of a gap in the growth of hereditary classes of ordered graphs. Furthermore, we prove a grid theorem which applies to all monadically NIP classes of structures (ordered or unordered), or equivalently, classes which do not transduce the class of all finite graphs.

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“…The twin-width invariant has been recently introduced [3,1,2]. Classes with bounded twinwidth include proper minor-closed classes and bounded rank-width graphs, and the property of having bounded twin-width is preserved by first-order transductions.…”

Section: Weakly χ P -Bounded Classes Of Graphsmentioning

confidence: 99%

From $χ$- to $χ_p$-bounded classes

Jiang¹,

Nešetřil²,

Mendez³

2020

Preprint

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χ-bounded classes are studied here in the context of star colorings and more generally χ pcolorings. This leads to natural extensions of the notion of bounded expansion class and to structural characterization of these. In this paper we solve two conjectures related to star coloring boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. We give structural characterizations of (strong and weak) χ p -bounded classes. On the way, we generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most ϕ(g, ω(G)) |G| holes of length g.

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Twin-width I: tractable FO model checking

Cited by 7 publications

References 0 publications

Stack-number is not bounded by queue-number

Stack-number is not bounded by queue-number

Ordered graphs of bounded twin-width

From $χ$- to $χ_p$-bounded classes

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