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Combinatorial optimization with 2-joins

Trotignon, N and Vuskovic, K (2012) Combinatorial optimization with 2-joins. Journal of Combinatorial Theory: Series B, 102 (1). 153 - 185 . ISSN 0095-8956

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Abstract

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset. We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.

Item Type: Article
Copyright, Publisher and Additional Information: © 2012, 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: Combinatorial optimization, Maximum clique, Minimum stable set, Coloring, Decomposition, Structure, 2-Join, Perfect graphs, Berge graphs, Even-hole-free 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: 18 Jun 2012 14:16
Last Modified: 24 Sep 2014 06:13
Published Version: http://dx.doi.org/10.1016/j.jctb.2011.06.002
Status: Published
Publisher: Elsevier
Identification Number: 10.1016/j.jctb.2011.06.002
URI: http://eprints.whiterose.ac.uk/id/eprint/74344

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