On the k-dominating set polytope of web graphs

Abstract

Abstract In this work we present some results on the polyhedral structure of the convex hull of integer points in polyhedra of the form { x ⩾ 0 : M x ⩾ k 1 } , for a 0, 1 matrix M and a positive integer number k. In particular, we consider the k-dominating set problem in a graph. Given a graph G = ( V , E ) , a set D ⊆ V is a k-dominating set if every vertex in V is adjacent to at least k vertices of D. The k-dominating set problem consists in finding a k-dominating set of minimum cardinality. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G and it is a natural generalization of the well-known dominating set polytope of a graph. We apply our results for general problems to the k-dominating set polytope of some particular families of web graphs.