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The stable set polytope of quasi-line graphs

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Abstract

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.

Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.

In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

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Authors and Affiliations

  1. Institute of Mathematics, University of Paderborn, Warburger Str. 100, Paderborn, 33098, Germany

    Friedrich Eisenbrand

  2. Dipartimento di Ingegneria dell’Impresa, Universitá di Tor Vergata, via del Politecnico 1, Roma, 00165, Italy

    Gianpaolo Oriolo

  3. Ecole Polytechnique Fédérale de Lausanne, SB-IMA-ROSO Station 8, Lausanne, CH-1015, Switzerland

    Gautier Stauffer

  4. Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” del CNR, Viale Manzoni 30, Roma, 00185, Italy

    Paolo Ventura

Authors
  1. Friedrich Eisenbrand
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  2. Gianpaolo Oriolo
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  3. Gautier Stauffer
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  4. Paolo Ventura
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Corresponding author

Correspondence to Friedrich Eisenbrand.

Additional information

An extended abstract of this work appeared in the proceedings of the Eleventh Conference on Integer Programming and Combinatorial, IPCO XI, 2005.

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Eisenbrand, F., Oriolo, G., Stauffer, G. et al. The stable set polytope of quasi-line graphs. Combinatorica 28, 45–67 (2008). https://doi.org/10.1007/s00493-008-2244-x

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  • DOI: https://doi.org/10.1007/s00493-008-2244-x

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