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On the global offensive alliance number of a graph

An offensive alliance in a graph $Γ=(V,E)$ is a set of vertices $S\subset V$ 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 $V\backslash S$, that is, $S$ is a dominating set of $Γ$. The offensive alliance number $a_o(Γ)$ (respectively, strong offensive alliance number $a_{\hat{o}}(Γ)$) is the minimum cardinality of an offensive (respectively, strong offensive) alliance in $Γ$. The global offensive alliance number $γ_o(Γ)$ and the global strong offensive alliance number $γ_{\hat{o}}(Γ)$ are defined similarly. Clearly, $a_o(Γ)\le γ_o(Γ)$ and $a_{\hat{o}}(Γ)\le γ_{\hat{o}}(Γ)$. It was shown in [Discuss. Math. Graph Theory, 24 (2004), no. 2, 263-275] that $ a_o(Γ)\le \frac{2n}{3}$ and $ a_{\hat{o}}(Γ)\le \frac{5n}{6}$, where $n$ denotes the order of $Γ$. In this paper we obtain several tight bounds on $γ_o(Γ)$ and $γ_{\hat{o}}(Γ)$ in terms of several parameters of $Γ$. For instance, we show that $\frac{2m+n}{3Δ+1} \le γ_o(Γ)\le \frac{2n}{3}$ and $\frac{2(m+n)}{3Δ+2} \leγ_{\hat{o}}(Γ)\le \frac{5n}{6}$, where $m$ denotes the size of $Γ$ and $Δ$ its maximum degree (the last upper bound holds true for all $Γ$ with minimum degree greatest or equal to two).