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Arie M.C.A. Koster (kosterzib.de) Abstract: Perfect graphs constitute a wellstudied graph class with a rich structure, reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs $G$ where the stable set polytope $STAB(G)$ coincides with the fractional stable set polytope $QSTAB(G)$. For all imperfect graphs $G$ it holds that $STAB(G) \subset QSTAB(G)$. It is, therefore, natural to use the difference between the two polytopes in order to decide how far an imperfect graph is away from being perfect; we discuss three different concepts, involving the facet set of $STAB( G)$, the disjunctive index of $QSTAB(G)$, and the dilation ratio of the two polytopes. Including only certain types of facets for $STAB(G)$, we obtain graphs that are in some sense close to perfect graphs, for example minimally imperfect graphs, and certain other classes of socalled rankperfect graphs. The imperfection ratio has been introduced by (Gerke and McDiarmid, 2001) as the dilation ratio of $STAB(G)$ and $QSTAB(G)$, whereas (Aguilera et al., 2003) suggest to take the disjunctive index of $Q STAB(G)$ as the imperfection index of $G$. For both invariants there exist no general upper bounds, but there are bounds known for the imperfection ratio of several graph classes (Coulonges et al. 2005, Gerke and McDiarmid, 2001). Outgoing from a graphtheoretical interpretation of the imperfection index, we conclude that the imperfection index is NPhard to compute and we prove that there exists no upper bound on the imperfect ion index for those graph classes with a known bounded imperfection ratio. Comparing the two invariants on those classes, it seems that the imperfection index measures imperfection much more roughly than the imperfection ratio; therefore, discuss possible directions for refinements. Keywords: perfect graphs, imperfection ratio, imperfection index Category 1: Combinatorial Optimization (Graphs and Matroids ) Category 2: Combinatorial Optimization (Polyhedra ) Citation: ZIBreport 0550 http://www.zib.de/ Download: [Compressed Postscript][PDF] Entry Submitted: 12/05/2005 Modify/Update this entry  
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