A New Perspective on FO Model Checking of Dense Graph Classes

Gajarský, Jakub; Hliněný, Petr; Óbdržálek, Jan; Lokshtanov, Daniel; Ramanujan, M. S.

doi:10.1145/2933575.2935314

Cited by 22 publications

(33 citation statements)

References 29 publications

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“…More generally, one may consider images under transductions of classes with forbidden minors or with forbidden topological minors. So far, suitable characterizations have been given for classes with structurally bounded degree [16] and with structurally bounded expansion [18]. Such characterizations, if e ciently constructive, are very helpful in the design of xed-parameter algorithms for the FO model-checking problem, as was done in the case of classes with structurally bounded degree [16].…”

Section: Discussionmentioning

confidence: 99%

“…So far, suitable characterizations have been given for classes with structurally bounded degree [16] and with structurally bounded expansion [18]. Such characterizations, if e ciently constructive, are very helpful in the design of xed-parameter algorithms for the FO model-checking problem, as was done in the case of classes with structurally bounded degree [16]. Based on the understanding revealed in [16,18], we hypothesize that such characterizations may rely on the concept of covers (see Section 4).…”

Section: Discussionmentioning

confidence: 99%

“…Such characterizations, if e ciently constructive, are very helpful in the design of xed-parameter algorithms for the FO model-checking problem, as was done in the case of classes with structurally bounded degree [16]. Based on the understanding revealed in [16,18], we hypothesize that such characterizations may rely on the concept of covers (see Section 4). For instance, transductions of classes with bounded expansion are characterized by the existence of such covers (see eorem 4.2).…”

Section: Discussionmentioning

confidence: 99%

“…Proof. If a class has structurally bounded expansion, then it has low shrubdepth covers [16], which are special instances of low rankwidth covers. Moreover, as bounded expansion classes are nowhere dense, they are monadically stable [1], hence structurally bounded expansion classes have a stable edge relation.…”

Section: Completing the Inductionmentioning

confidence: 99%

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Linear rankwidth meets stability

Nešetřil¹,

Mendez²,

Rabinovich³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSOtransductions of paths. These results show a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes: 1) Graphs with linear rankwidth at most r are linearly χ-bounded. Actually, they have bounded c-chromatic number, meaning that they can be colored with f (r) colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary family F of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in F . 3) For a class C with bounded linear rankwidth the following conditions are equivalent: a) C is stable, b) C excludes some half-graph as a semi-induced subgraph, c) C is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Completing the Inductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear rankwidth meets stability

Nešetřil¹,

Mendez²,

Rabinovich³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…One line of research tries to extend tractability to even more general or different classes of graphs and structures, see e.g. [17,18,19].…”

Section: Introductionmentioning

confidence: 99%

Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

Heuvel

Kreutzer

Pilipczuk

et al. 2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula.It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [13], excludes a fixed minor [11] or a fixed topological minor [10,25]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.

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A Generalization of the Łoś-Tarski Preservation Theorem over Classes of Finite Structures

Sankaran

Adsul

Chakraborty

2014

Mathematical Foundations of Computer Science 2014

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We investigate a generalization of the Loś-Tarski preservation theorem via the semantic notion of preservation under substructures modulo k-sized cores. It was shown earlier that over arbitrary structures, this semantic notion for first-order logic corresponds to definability by ∃ k ∀ * sentences. In this paper, we identify two properties of classes of finite structures that ensure the above correspondence. The first is based on well-quasi-ordering under the embedding relation. The second is a logic-based combinatorial property that strictly generalizes the first. We show that starting with classes satisfying any of these properties, the classes obtained by applying operations like disjoint union, cartesian and tensor products, or by forming words and trees over the classes, inherit the same property. As a fallout, we obtain interesting classes of structures over which an effective version of the Loś-Tarski theorem holds.

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A New Perspective on FO Model Checking of Dense Graph Classes

Cited by 22 publications

References 29 publications

Linear rankwidth meets stability

Linear rankwidth meets stability

Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

A Generalization of the Łoś-Tarski Preservation Theorem over Classes of Finite Structures

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