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Lovász-Schrijver PSD-Operator on Some Graph Classes Defined by Clique Cutsets

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Combinatorial Optimization (ISCO 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10856))

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Abstract

This work is devoted to the study of the Lovász-Schrijver PSD-operator \(LS_+\) applied to the edge relaxation \(\mathrm{ESTAB}(G)\) of the stable set polytope \(\mathrm{STAB}(G)\) of a graph G. In order to characterize the graphs G for which \(\mathrm{STAB}(G)\) is achieved in one iteration of the \(LS_+\)-operator, called \(LS_+\)-perfect graphs, an according conjecture has been recently formulated (\(LS_+\)-Perfect Graph Conjecture). Here we study two graph classes defined by clique cutsets (pseudothreshold graphs and graphs without certain Truemper configurations). We completely describe the facets of the stable set polytope for such graphs, which enables us to show that one class is a subclass of \(LS_+\)-perfect graphs, and to verify the \(LS_+\)-Perfect Graph Conjecture for the other class.

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Notes

  1. 1.

    An anticomponent is an inclusion-wise maximal subgraph \(G'\) of G such that \(\overline{G}'\) is a component of \(\overline{G}\).

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Authors and Affiliations

  1. LIMOS (UMR 6158 CNRS), University Clermont Auvergne, Clermont-Ferrand, France

    Annegret Wagler

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  1. Annegret Wagler
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Correspondence to Annegret Wagler .

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Editors and Affiliations

  1. Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA

    Jon Lee

  2. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI), CNR, Rome, Italy

    Giovanni Rinaldi

  3. LAMSADE, Université Paris-Dauphine, Paris, France

    A. Ridha Mahjoub

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Wagler, A. (2018). Lovász-Schrijver PSD-Operator on Some Graph Classes Defined by Clique Cutsets. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_35

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  • DOI: https://doi.org/10.1007/978-3-319-96151-4_35

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  • Print ISBN: 978-3-319-96150-7

  • Online ISBN: 978-3-319-96151-4

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