Optimization of Dispersed Generation Systems including Risk Aversion

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 vʹs 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 ) (respectively, global strong offensive alliance number γ o ˆ ( Γ ) ) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in Γ. In this paper we obtain several tight bounds on γ o ( Γ ) and γ o ˆ ( Γ ) in terms of several parameters of Γ.