A Polyhedral View to Generalized Multiple Domination and Limited Packing

Abstract

Given an undirected simple graph $G=(V,E)$ and integer values $f_v, v\in V$, a node subset $D\subseteq V$ is called an f-tuple dominating set if, for each node $v\in V$, its closed neighborhood intersects D in at least $f_v$ nodes. We investigate the polyhedral structure of the polytope that is defined as the convex hull of the incidence vectors in $\mathbb {R}^{V}$ of the f-tuple dominating sets in G. We provide a complete formulation for the case of stars and introduce a new family of (generally exponentially many) inequalities which are valid for the f-tuple dominating set polytope and that can be separated in polynomial time. A corollary of our results is a proof that a conjecture present in the literature on a complete formulation of the 2-tuple dominating set polytope of trees does not hold. Investigations on adjacency properties in the 1-skeleton of the f-tuple dominating set polytope are also reported.