Offensive rr-alliances in graphs

Let G=(V,E)G=(V,E) be a simple graph. For a nonempty set X⊂VX⊂V, and a vertex v∈V,δX(v)v∈V,δX(v) denotes the number of neighbors vv has in XX. A nonempty set S⊂VS⊂V is an offensive  rr-alliance   in GG if δS(v)≥δS̄(v)+r,∀v∈∂(S), where ∂(S)∂(S) denotes the boundary of SS. An offensive rr-alliance SS is called global if it forms a dominating set. The global offensive  rr-alliance number   of GG, denoted by γro(G), is the minimum cardinality of a global offensive rr-alliance in GG. We show that the problem of finding optimal (global) offensive rr-alliances is NP-complete and we obtain several tight bounds on γro(G).