关于图的全局进攻联盟数
On the global offensive alliance number of a graph
摘要：
图$Γ=(V,E)$中的进攻联盟是一组顶点$S\subset V$，其中对于其边界上的每个顶点$v$，它认为$v$的闭邻域中的大多数顶点都在$S$中。在强大的进攻联盟的情况下，需要严格的多数。如果$S$影响到$V\backslash S$中的每一个顶点，即$S$是$Γ$的支配集，则称该联盟是全局的。进攻联盟数$a_o(Γ)$（分别为强进攻联盟数$a_{\hat{o}}(Γ)$）是$Γ$中进攻（分别为强进攻）联盟的最小基数。全局进攻联盟号$γ_o(Γ)$和全局强进攻联盟号$γ_{\hat{o}}(Γ)$被类似地定义。显然，$a_o(Γ)\le γ_o(Γ)$和$a_{\hat{o}}(Γ)\le γ_{\hat{o}}(Γ)$。它在[讨论]中显示。Math. GraphTheory，24（2004），no.2，263-275]，$ a_o(Γ)\le \frac{2n}{3}$和$ a_{\hat{o}}(Γ)\le \frac{5n}{6}$，其中$n$表示$Γ$的阶。本文利用$Γ$的几个参数得到了$γ_o(Γ)$和$γ_{\hat{o}}(Γ)$的几个紧界.例如，我们证明了$\frac{2m+n}{3Δ+1} \le γ_o(Γ)\le \frac{2n}{3}$和$\frac{2(m+n)}{3Δ+2} \leγ_{\hat{o}}(Γ)\le \frac{5n}{6}$，其中$m$表示$Γ$的大小，$Δ$表示其最大程度（最后一个上限适用于所有最小程度最大或等于2的$Γ$）。
英文摘要：
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).
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