## The extreme points of QSTAB(G) and its implications

Please always quote using this URN: urn:nbn:de:0297-zib-9249

- Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations w.r.t different concepts. Perfect graphs are, e.g., characterized as precisely those graphs $G$ where the stable set polytope STAB$(G)$ coincides with the clique constraint stable set polytope QSTAB$(G)$. For all imperfect graphs STAB$(G) \subset$ QSTAB$(G)$ holds and, therefore, it is natural to measure imperfection in terms of the difference between STAB$(G)$ and QSTAB$(G)$. Several concepts have been developed in this direction, for instance the dilation ratio of STAB$(G)$ and QSTAB$(G)$ which is equivalent to the imperfection ratio imp$(G)$ of $G$. To determine imp$(G)$, both knowledge on the facets of STAB$(G)$ and the extreme points of QSTAB$(G)$ is required. The anti-blocking theory of polyhedra yields all {\em dominating} extreme points of QSTAB$(G)$, provided a complete description of the facets of STAB$(\overline G)$ is known. As this is typically not the case, we extend the result on anti-blocking polyhedra to a {\em complete} characterization of the extreme points of QSTAB$(G)$ by establishing a 1-1 correspondence to the facet-defining subgraphs of $\overline G$. We discuss several consequences, in particular, we give alternative proofs of several famous results.

Author: | Arie M.C.A. Koster, Annegret K. Wagler |
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Document Type: | ZIB-Report |

Tag: | imperfection ratio; perfect graphs; stable set polytope |

MSC-Classification: | 05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C17 Perfect graphs |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut | |

Date of first Publication: | 2006/06/01 |

Series (Serial Number): | ZIB-Report (06-30) |

Published in: | An extended abstract appeared under the title "On Determining the Imperfection Ratio" in: Electronic Notes in Discrete Mathematics 25 (2006) 177-181 |