Some new hereditary classes where graph coloring remains NP-hard

Schindl, David

doi:10.1016/j.disc.2005.03.003

Cited by 26 publications

(21 citation statements)

References 8 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: 98%

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: 98%

Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

Dabrowski

Golovach

Paulusma

2013

Graph-Theoretic Concepts in Computer Science

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“…Some initial results have been obtained by Král' et al [15], Schindl [22] and a number of authors studying the H-free graphs, in which one of the two graphs in H is the triangle [5,7,13,19]. Another extension is to classify the computational complexity of k-Coloring and other variants of coloring for H-free graphs where k is a fixed integer and H is a fixed graph.…”

Section: Theorem 1 ([15])mentioning

confidence: 99%

List Coloring in the Absence of a Linear Forest

Couturier¹,

Golovach

Kratsch³

et al. 2013

Algorithmica

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The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-013-9777-0.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. Abstract. The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1, . . . , k}.Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k-Coloring can be solved in polynomial time for graphs with no induced rP1 + P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1 + P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H.

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“…This work has been extended by Schindl [23]. Maffray and Preissmann [20] showed that Coloring is NP-complete for (C 3 , K 1,5 )-free graphs.…”

Section: Theorem 1 ([17])mentioning

confidence: 99%