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

“…Malyshev and Lobanova proved that, for all , Coloring is polynomial‐time solvable for ‐free graphs, which generalizes an earlier result of Dabrowski et al. for the class of ‐free graphs. 11.The class of ‐free graphs has bounded clique width , hence we apply Theorem (i). …”

“…">10.Dabrowski et al. proved that for every two integers and , Coloring is polynomial‐time solvable for ‐free graphs. Malyshev and Lobanova proved that, for all , Coloring is polynomial‐time solvable for ‐free graphs, which generalizes an earlier result of Dabrowski et al.…”

Two cases of polynomial-time solvability for the coloring problem

“…A study for forbidden pairs was also initialized in the paper. Only partial results are known in the case of two forbidden induced subgraphs (Dabrowski et al 2012(Dabrowski et al , 2014Golovach and Paulusma 2013;Kral' et al 2001;Korpeilainen et al 2011;Lozin and Malyshev 2014;Schindl 2005). Moreover, a complete classification for pairs is wide open.…”

Hereditary graph classes: When the complexities of coloring and clique cover coincide

“…Both polynomial‐time and ‐completeness results are known for ‐free graphs, as shown by many teams of researchers (see e.g. ), but the complexity classification is far from complete: even if we forbid two graphs and of up to four vertices, there are still three open cases left, namely when ; see (the graph is the claw and is the ‐vertex cycle). We refer to [18, Theorem 21] for a summary and to the recent paper , in which the number of missing cases when and are both connected graphs on at most 5 vertices was reduced from 10 to 8.…”

“…Hence, the class of ‐free graphs is finite, which means that coloring is constant‐time solvable for this graph class. Dabrowski et al researched the effect on the complexity of changing the forbidden subgraph by adding an extra edge and proved that coloring is polynomial‐time solvable on ‐free graphs for every integer . A result of Malyshev implies that coloring is polynomial‐time solvable for ‐free graphs.…”