A Characterization of b‐Perfect Graphs

Hoàng, Chı́nh T.; Maffray, Frédéric; Mechebbek, Meriem

doi:10.1002/jgt.20635

Cited by 19 publications

(8 citation statements)

References 11 publications

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“…More results on Colouring for (H 1 , H 2 )-free graphs can be found in a number of other papers [4,5,11,20,24,28,33,34,36], all of which are summarized in Theorem 3 given below, together with the above results and a weaker formulation of our new result (Statement (ii)-8). In this theorem, the graph C denotes the bull, which is the graph with vertices a, b, c, d, e and edges ab, ac, ad, bc, be and the graph C * 3 denotes the hammer, which is the graph with vertices a, b, c, d, e and edges ab, ac, ad, bc, de.…”

Section: Theorem 2 ([17]mentioning

confidence: 99%

Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

Dabrowski

Golovach

Paulusma

2013

Graph-Theoretic Concepts in Computer Science

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Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Theoretical computer science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Theoretical computer science, 522, 2014, 10.1016/j.tcs.2013.12.004 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free graphs with |H| = 1 has been completely classified. When |H| = 2, the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H1, H2}, where we allow H1 to have a single edge and H2 to have a single nonedge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is fixed-parameter tractable when parameterized by |H1| + |H2|. As a by-product, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.

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Section: Theorem 2 ([17]mentioning

confidence: 99%

Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

Dabrowski

Golovach

Paulusma

2013

Graph-Theoretic Concepts in Computer Science

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“…Lozin and Malyshev [28] showed that Coloring is polynomial-time solvable for (K 1,3 , 2P 2 )-free graphs. Hoàng, Maffray and Mechebbek [19] characterised the so-called b-perfect graphs in terms of forbidden induced subgraphs and showed that Coloring is polynomial-time solvable for (2P 2 , P 5 )-free graphs. Using their characterisation one can show that (2P 2 , P 5 )-free graphs are b-perfect.…”

Section: Related Workmentioning

confidence: 99%

List Coloring in the Absence of Two Subgraphs

Golovach

Paulusma

2013

Lecture Notes in Computer Science

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Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Discrete applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Discrete applied mathematics, 166, 2014, 10.1016/j.dam.2013.10.010 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The List Coloring problem is that of testing whether a given graph G = (V, E) has a coloring c that respects a given list assignment L, i.e., whether G has a mapping c :If a graph G has no induced subgraph isomorphic to some graph of a pair {H1, H2}, then G is called (H1, H2)-free. We completely characterize the complexity of List Coloring for (H1, H2)-free graphs.

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“…Note that b-perfect graphs are b-monotonous. Recently, Hoang et al have characterized the b-perfect graphs by forbidden subgraphs [13]. For other results on b-coloring one can see [10,7,12] [15] or [9].…”

Section: Introductionmentioning

confidence: 99%

On quasi-monotonous graphs

Kouider

2016

Discrete Applied Mathematics

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a b s t r a c tA dominating coloring by k colors is a proper k coloring where every color i has a representative vertex x i adjacent to at least one vertex in each of the other classes. The b-chromatic number, b(G), of a graph G is the largest integer k such that G admits a dominating coloring by k colors.for every induced subgraph H 1 of G and every subgraph H 2 of H 1 . Here we say that a graph G is quasi b-monotonous, or simply quasi-monotonous, if for every vertexWe shall study the quasi-monotonicity of several classes. We show in particular that chordal graphs are not quasi-monotonous in general, whereas chordal graphs with large b-chromatic number, and (P, coP, chair, diamond)-free graphs are quasimonotonous. The (P 5 , P, dart)-free graphs are monotonous. Finally we give new bounds for the b-chromatic number of any vertex deleted subgraph of a chordal graph.

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A Characterization of b‐Perfect Graphs

Cited by 19 publications

References 11 publications

Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

List Coloring in the Absence of Two Subgraphs

On quasi-monotonous graphs

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