On f-domination: polyhedral and algorithmic results

Abstract

Given an undirected simple graph G with node set V and edge set E, let $f_v$, for each node $v \in V$, 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 $v\in V{{\setminus }}D$ 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 $c_v\in {\mathbb {R}}$ for each node $v\in V$, find an f-dominating set D in G whose cost, given by $\sum _{v\in D}{c_v}$, is a minimum.