Colouring vertices of triangle-free graphs without forests

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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“…Combining this observation with such existing results for k-Colouring [7,11,19] gives us a number of polynomialtime solvable cases [13].…”

Section: Theorem 2 ([17]mentioning

confidence: 87%

Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

Dabrowski

Golovach

Paulusma

2013

Graph-Theoretic Concepts in Computer Science

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“…This shows Case (ii):5, whereas we obtain Case (ii):4 by using the same arguments together with a result of Král' et al [24], who showed that for any fixed graph H 2 , Coloring is polynomialtime solvable on (C 3 , H 2 )-free graphs if and only if it is so for (C + 3 , H 2 )-free graphs. Case (ii):3 is proven by combining the latter result with corresponding results of Dabrowski, Lozin, Raman and Ries [11] for (C 3 , H 2 )-free graphs that are obtained by combining a number of new results with known results for H 2 = K 1,4 [24], H 2 = S 1,2,2 [32], H 2 = P 2 + P 4 [6], H 2 = 2P 3 [7], H 2 = P 6 [3], H 2 is the cross [33] (the graph obtained from K 1,4 by making a new vertex adjacent to one of its leafs) and H 2 is the 'H'-graph [32] (the graph obtained from K 1,3 by making two new non-adjacent vertices adjacent to the same leaf). Finally, Case (ii):8 has been shown by Dabrowski, Golovach and Paulusma [10].…”

Section: Related Workmentioning

confidence: 78%

“…The latter is true for F = sP 1 + P 5 [8] and F = sP 2 (see e.g. [11]). This shows Case (ii):5, whereas we obtain Case (ii):4 by using the same arguments together with a result of Král' et al [24], who showed that for any fixed graph H 2 , Coloring is polynomialtime solvable on (C 3 , H 2 )-free graphs if and only if it is so for (C + 3 , H 2 )-free graphs.…”

Section: Related Workmentioning

confidence: 99%

List coloring in the absence of two subgraphs

Golovach

Paulusma

2014

Discrete Applied Mathematics

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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 re ected in this document. Changes may have been made to this work since it was submitted for publication. A de nitive 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-pro t 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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“…Maffray and Preissmann [20] showed that Coloring is NP-complete for (C 3 , K 1,5 )-free graphs. Broersma et al [5] showed that Coloring is polynomial-time solvable for (C 3 , 2P 3 )-free graphs, hereby completing a study of Dabrowski et al [9] who considered the Coloring problem restricted to (C 3 , H)-free graphs for graphs H on at most six vertices.…”

Section: Theorem 1 ([17])mentioning

confidence: 99%