FO model checking on geometric graphs

Hliněný, Petr; Pokrývka, Filip; Roy, Bishnupada

doi:10.1016/j.comgeo.2018.10.001

Cited by 14 publications

(13 citation statements)

References 36 publications

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“…See e.g. [17,7,19,21,4,9,10,27,8,22,28,16,5,6,13,20]. One of the main goals of this line of research is to find a structural property P which precisely defines those graph classes C for which model checking of first-order logic is tractable.…”

Section: Introductionmentioning

confidence: 99%

First-Order Interpretations of Bounded Expansion Classes

Gajarský

Kreutzer

Nešetřil

et al. 2020

ACM Trans. Comput. Logic

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The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

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Section: Introductionmentioning

confidence: 99%

First-Order Interpretations of Bounded Expansion Classes

Gajarský

Kreutzer

Nešetřil

et al. 2020

ACM Trans. Comput. Logic

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“…Last to say, the results of this paper offer us new possibilities to find the exact boundary between bounded and unbounded twin-width (and perhaps between efficient and intractable FO model checking) in the class of interval graphs, and hence finish the line of research started (before the birth of twin-width) in [10,11].…”

Section: Discussionmentioning

confidence: 89%

Twin-width and Transductions of Proper k-Mixed-Thin Graphs

Balabán¹,

Hliněný²,

Jedelský³

2022

Preprint

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The new graph parameter twin-width, recently introduced by Bonnet, Kim, Thomassé and Watrigant, admits an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the algorithmic point of view. In particular, such classes (of small twin-width) include proper interval graphs, and (as digraphs) posets of width k. Inspired by an existing generalization of interval graphs into so-called k-thin graphs, we define a new class of proper k-mixed-thin graphs which largely generalizes proper interval graphs. We prove that proper k-mixed-thin graphs have twin-width linear in k, and that a certain subclass of k-mixed-thin graphs is transductionequivalent to posets of width k such that there is a quadratic relation between k and k .

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“…Our refined results on twin-width of posets can bring faster theoretical algorithms for FO model checking of posets [7] and of other classes which have previously been reduced to posets of bounded width, such as [8,6,9].…”

Section: Introductionmentioning

confidence: 89%

Twin-Width is Linear in the Poset Width

Balabán¹,

Hliněný²

2021

Preprint

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Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g ṫo digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomassé and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation from poset width to twin-width was indirect and giving only a very loose doubleexponential bound. We prove that posets of width d have twin-width at most 9d with a direct and elegant argument, and show that this bound is asymptotically tight. Specially, for posets of width 2 we prove that in the worst case their twin-width is also equal 2. These two theoretical results are complemented with straightforward algorithms to construct the respective contraction sequence for a given poset.

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FO model checking on geometric graphs

Cited by 14 publications

References 36 publications

First-Order Interpretations of Bounded Expansion Classes

First-Order Interpretations of Bounded Expansion Classes

Twin-width and Transductions of Proper k-Mixed-Thin Graphs

Twin-Width is Linear in the Poset Width

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