A polyhedral view to a generalization of multiple domination

Given an undirected simple graph G=(V,E) and integer values f_v,v∈V, a node subset D⊆V is called an f-tuple dominating set if, for each node v∈V, its closed neighborhood intersects D in at least f_v nodes. We study the polytope that is defined as the convex hull of the incidence vectors in R^V of the f-tuple dominating sets in G. New families of valid inequalities are introduced and a complete formulation is given for the case of stars. A corollary of our results is a proof that the conjecture reported in Argiroffo (2013) on a complete formulation of the 2-tuple dominating set polytope of trees does not hold. Preliminary computational results are also reported.