On the global offensive alliance number of a graph

An offensive alliance in a graph @C=(V,E) is a set of vertices [email protected]?V where for each vertex v in its boundary 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 [email protected]?S, that is, S is a dominating set of @C. The global offensive alliance [email protected]"o(@C) is the minimum cardinality of a global offensive alliance in @C. An offensive alliance is connected if its induced subgraph is connected. The global-connected offensive alliance number, @c"c"o(@C), is the minimum cardinality of a global-connected offensive alliance in @C. In this paper we obtain several tight bounds on @c"o(@C) and @c"c"o(@C) in terms of several parameters of @C. The case of strong alliances is studied by analogy.