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.