Colouring of graphs with Ramsey-type forbidden subgraphs

Abstract

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.

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

“…A (partial) proof of Theorem 21 can be found in the papers of Golovach and Paulusma [44] and Dabrowski et al [23]. Note that, by symmetry, the graphs H 1 and H 2 may be swapped in each of the subcases of Theorem 21.…”

“…Dabrowski et al [23] proved that for every two integers s ≥ 0 and t ≥ 0, COL-ORING is polynomial-time solvable for (sP 1 + P 2 , tP 1 + P 2 )-free graphs. Malyshev and Lobanova [87] proved that, for all t ≥ 0, COLORING is polynomialtime solvable for (P 5 , tP 1 + P 2 )-free graphs, which generalizes an earlier result of Dabrowski et al [23] for the class of (2P 2 , tP 1 + P 2 )-free graphs. 11.…”

Classifying the clique-width of H-free bipartite graphs

“…In particular, many papers have determined the clique-width of graph classes characterized by one or more forbidden induced subgraphs [1,2,[5][6][7][8][9][10][11][12]15,20,[23][24][25][26].…”

List coloring in the absence of two subgraphs

“…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].…”