Global offensive alliances in graphs

An offensive alliance in a graph Γ=(V,E) is a set of vertices S⊂V where for every vertex v in its boundary it holds that the majority of vertices in v's closed neighborhood are in S. In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V\S, that is, S is a dominating set of Γ. The global offensive alliance number γo(Γ) (respectively, global strong offensive alliance number ) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in Γ. In this paper we obtain several tight bounds on γo(Γ) and in terms of several parameters of Γ.