Stability critical graphs and ranks facets of the stable set polytope

Abstract

Rank inequalities due to stability critical ( a-critical) graphs are used to develop a finite nested sequence of linear relaxations of the stable set polytope, the strongest of which provides an integral max-min relation: In a simple graph, the maximum size of a stable set is equal to the minimum (weighted) value of a cover of nodes by a-critical subgraphs. For a simple graph containing no even subdivision of K 4, these results imply that every rank facet is due either to an edge or to an odd cycle; consequently, the max-min relation specializes to give that the cardinality of a largest stable set equals the minimum value of a node covering by edges and odd cycles. This leads to a polynomial-time algorithm to find a maximum stable set and a minimum valued cover of nodes by edges and odd cycles in such a graph.