On the global offensive alliance number of a graph

Abstract

An offensive alliance in a graph Γ = ( V , E ) is a set of vertices S ⊂ 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 V ∖ S , that is, S is a dominating set of Γ . The global offensive alliance number γ o ( Γ ) is the minimum cardinality of a global offensive alliance in Γ . An offensive alliance is connected if its induced subgraph is connected. The global-connected offensive alliance number, γ c o ( Γ ) , is the minimum cardinality of a global-connected offensive alliance in Γ . In this paper we obtain several tight bounds on γ o ( Γ ) and γ c o ( Γ ) in terms of several parameters of Γ . The case of strong alliances is studied by analogy.