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Colouring of graphs with Ramsey-type forbidden subgraphs.

Dabrowski, K.K. and Golovach, P.A. and Paulusma, D. (2013) 'Colouring of graphs with Ramsey-type forbidden subgraphs.', in Graph-theoretic concepts in computer science : 39th International Workshop, WG 2013, 19-21 June 2013, Lübeck, Germany ; revised papers. Berlin, Heidelberg: Springer, pp. 201-212. Lecture notes in computer science. (8165).

Abstract

A colouring of a graph G = (V;E) is a mapping c : V ! f1; 2; : : :g such that c(u) 6= c(v) if uv 2 E; if jc(V )j k then c is a k-colouring. 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 jHj = 1 has been completely classied. When jHj = 2, the classication is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs fH1;H2g, 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 xed-parameter tractable when parameterized by jH1j + jH2j. As a byproduct, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.

Item Type:Book chapter
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1007/978-3-642-45043-3_18
Publisher statement:The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-45043-3_18
Date accepted:No date available
Date deposited:14 January 2015
Date of first online publication:2013
Date first made open access:No date available

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