Offensive r-alliances in graphs

Let G = (V , E) be a simple graph. For a nonempty set X ⊂ V , and a vertex v ∈ V , δX (v) denotes the number of neighbors v has in X . A nonempty set S ⊂ V is an offensive r-alliance in G if δS(v) ≥ δS̄(v)+ r,∀v ∈ ∂(S), where ∂(S) denotes the boundary of S. An offensive ralliance S is called global if it forms a dominating set. The global offensive r-alliance number of G, denoted by γ o r (G), is the minimum cardinality of a global offensive r-alliance in G. We show that the problem of finding optimal (global) offensive r-alliances is NP-complete and we obtain several tight bounds on γ o r (G). © 2008 Elsevier B.V. All rights reserved.