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Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences

Kloks, T, Muller, H and Vuskovic, K (2009) Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences. Journal of Combinatorial Theory: Series B, 99 (5). 733 - 800 . ISSN 0095-8956

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Abstract

In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e., chordless cycles of even length) and diamonds (i.e., a graph obtained from a clique of size 4 by removing an edge). We say that such graphs are (even-hole, diamond)-free. For this class of graphs we first obtain a decomposition theorem, using clique cutsets, bisimplicial cutsets (which is a special type of a star cutset) and 2-joins. This decomposition theorem is then used to prove that every graph that is (even-hole, diamond)-free contains a simplicial extreme (i.e., a vertex that is either of degree 2 or whose neighborhood induces a clique). This characterization implies that for every (even-hole, diamond)-free graph G, χ(G)⩽ω(G)+1 (where χ denotes the chromatic number and ω the size of a largest clique). In other words, the class of (even-hole, diamond)-free graphs is a χ-bounded family of graphs with the Vizing bound for the chromatic number. The existence of simplicial extremes also shows that (even-hole, diamond)-free graphs are β-perfect, which implies a polynomial time coloring algorithm, by coloring greedily on a particular, easily constructable, ordering of vertices. Note that the class of (even-hole, diamond)-free graphs can also be recognized in polynomial time.

Item Type: Article
Copyright, Publisher and Additional Information: © 2009, Elsevier. This is an author produced version of a paper published in Journal of Combinatorial Theory: Series B. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: even-hole-free graphs, decomposition, chi-bounded families, beta-perfect graphs, greedy coloring algorithm, beta-perfect graphs
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 20 Jun 2012 13:07
Last Modified: 09 Jun 2014 18:40
Published Version: http://dx.doi.org/10.1016/j.jctb.2008.12.005
Status: Published
Publisher: Elsevier
Identification Number: 10.1016/j.jctb.2008.12.005
URI: http://eprints.whiterose.ac.uk/id/eprint/74350

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