Decomposition techniques applied to the Clique-Stable set separation problem

In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K∩S=∅, there exists a cut (W,W′) in C such that K⊆W and S⊆W′. Starting from a question concerning extended formulations of the Stab...

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Published inDiscrete mathematics Vol. 341; no. 5; pp. 1492 - 1501
Main Authors Bousquet, Nicolas, Lagoutte, Aurélie, Maffray, Frédéric, Pastor, Lucas
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.05.2018
Elsevier
Subjects
Apple-free graphs
Clique-Stable set separation
Combinatorics
Computer Science
Discrete Mathematics
Extended formulation
Graph decomposition
Mathematics
Stable set polytope
Apple-free graphs
Graph decomposition
Stable set polytope
Clique-Stable set separation
Extended formulation
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Summary:In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K∩S=∅, there exists a cut (W,W′) in C such that K⊆W and S⊆W′. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis (Yannakakis, 1991) asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive and negative results related to this question were given recently. Here we show how graph decomposition can be used to prove that a class of graphs admits a polynomial CS-separator. We apply this method to apple-free graphs and cap-free graphs.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2017.10.014