On f-domination: polyhedral and algorithmic results

Given an undirected simple graph G with node set V and edge set E, let f_v, for each node v∈ V, denote a nonnegative integer value that is lower than or equal to the degree of v in G. An f-dominating set in G is a node subset D such that for each node v∈ V\ D at least f_v of its neighbors belong to D. In this paper, we study the polyhedral structure of the polytope defined as the convex hull of all the incidence vectors of f-dominating sets in G and give a complete description for the case of trees. We prove that the corresponding separation problem can be solved in polynomial time. In addition, we present a linear-time algorithm to solve the weighted version of the problem on trees: Given a cost c_v∈ R for each node v∈ V, find an f-dominating set D in G whose cost, given by ∑ _{v ∈ D}c_v, is a minimum.