Cliquewidth III: The Odd Case of Graph Coloring Parameterized by Cliquewidth

Golovach, Petr A.; Lokshtanov, Daniel; Saurabh, Saket; Zehavi, Meirav

doi:10.1137/1.9781611975031.19

Cited by 13 publications

(15 citation statements)

References 47 publications

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“…ETH fails [21]. However, it follows from what precedes that for each class C with bounded rankwidth and stable edge relation there is an O(n 3 )-time algorithm, which gives a constant factor approximation for the chromatic number.…”

Section: Completing the Inductionmentioning

confidence: 99%

Rankwidth meets stability

Nešetřil¹,

Mendez²,

Pilipczuk³

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs C is monadically stable if it is impossible to de ne arbitrarily long linear orders in vertex-colored graphs from C using a xed rst-order formula. Similarly, monadic dependence corresponds to the impossibility of de ning all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. us, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, modeltheoretical context. We explore this emerging theory by proving the following: 1) A class of graphs C is a rst-order transduction of a class with bounded treewidth if and only if C has bounded rankwidth and a stable edge relation (i.e. graphs from C exclude some half-graph as a semi-induced subgraph). 2) If a class of graphs C is monadically dependent and not monadically stable, then C has in fact an unstable edge relation.As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly χ-bounded. Our proofs are e ective and lead to polynomial time algorithms. * is paper is a part of projects that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No 810115 -D , and No 677651 -T ).

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Section: Completing the Inductionmentioning

confidence: 99%

Rankwidth meets stability

Nešetřil¹,

Mendez²,

Pilipczuk³

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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“…It is known that the chromatic number of graphs with (linear) cliquewidth at most k cannot be computed in f (k)n 2 o (k) time for any computable function f , unless ETH fails [21]. However, it follows from what precedes that for each class C with bounded rankwidth and stable edge relation there is an O(n 3 )-time algorithm, which gives a constant factor approximation for the chromatic number.…”

Section: Completing the Inductionmentioning

confidence: 99%

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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“…For other graph problems where boundedness of clique-width is used to classify their computational complexity on hereditary graph classes, see, for example, [28,44]. We refer to [11,41,79,80,89] for parameterized complexity results on clique-width.…”

Section: Algorithmic Consequencesmentioning

confidence: 99%

Clique-width for hereditary graph classes

Dabrowski¹,

Johnson²,

Paulusma³

2019

Surveys in Combinatorics 2019

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Cliquewidth III: The Odd Case of Graph Coloring Parameterized by Cliquewidth

Cited by 13 publications

References 47 publications

Rankwidth meets stability

Rankwidth meets stability

Linear rankwidth meets stability

Clique-width for hereditary graph classes

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