A class of h-perfect graphs

Abstract

A graph is said to be h-perfect if the convex hull of its independent sets is defined by the constraints corresponding to cliques and odd holes, and the nonnegativity constraints. Series-parallel graphs and perfect graphs are h-perfect. The purpose of this paper is to extend the class of graphs known to be h-perfect. Thus, given a graph which is the union of a bipartite graph G 1 and a graph G 2 having exactly two common nodes a and b, and no edge in common, we prove that G is h-perfect if so is the graph obtained from G by replacing G 1 by an a-b chain (the length of which depends on G 1). This result enables us to prove that the graph obtained by substituting bipartite graphs for edges of a series-parallel graph is h-perfect, and also that the identification of two nodes of a bipartite graph yields an h-perfect graph (modulo a reduction which preserves h-perfection).