On the global offensive alliance number of a graph

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 offensive alliance number ao(Γ) (respectively, strong offensive alliance number aô(Γ)) is the minimum cardinality of an offensive (respectively, strong offensive) alliance in Γ. The global offensive alliance number γo(Γ) and the global strong offensive alliance number γô(Γ) are defined similarly. Clearly, ao(Γ) ≤ γo(Γ) and aô(Γ) ≤ γô(Γ). It was shown in [Discuss. Math. Graph Theory 24 (2004), no. 2, 263-275] that ao(Γ) ≤ 2n 3 and aô(Γ) ≤ 5n 6 , where n denotes the order of Γ. In this paper we obtain several tight bounds on γo(Γ) and γô(Γ) in terms of several parameters of Γ. For instance, we show that 2m+n 3∆+1 ≤ γo(Γ) ≤ 2n 3 and 2(m+n) 3∆+2 ≤ γô(Γ) ≤ 5n 6 , where m denotes the e-mail:[email protected] e-mail:[email protected] 1 size of Γ and ∆ its maximum degree (the last upper bound holds true for all Γ with minimum degree greatest or equal to two).