三次图中的进攻联盟
Offensive alliances in cubic graphs
摘要：
图$Γ=(V,E)$中的进攻联盟是一组顶点$S\subset V$，其中对于其边界中的每个顶点$v$，它保持$v$的闭邻域中的大多数顶点在$S$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。全球进攻联盟号$γ_o(Γ)$（分别为全球强进攻联盟号$γ_{\hat{o}}(Γ)$）是PDMATHAMR中全球进攻（分别为全球强进攻）联盟的最小基数。如果$Γ$具有全局独立进攻联盟，则\emph{global independent offensive alliance number} $γ_i(Γ)$是$Γ$的所有独立全局进攻联盟中的最小基数。本文研究了三次图的全局（强）联盟数的数学性质。例如，我们证明了对所有连通三次图的$n$，$$\frac{2n}{5}\le γ_i(Γ)\le \frac{n}{2}\le γ_{\hat{o}}(Γ)\le \frac{3n}{4} \le γ_{\hat{o}}({\cal L}(Γ))=γ_{o}({\cal L}(Γ))\le n,$$，其中${\cal L}(Γ)$表示$Γ$的线图.所有上述边界都是紧的。
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 global offensive alliance number $γ_o(Γ)$ (respectively, global strong offensive alliance number $γ_{\hat{o}}(Γ)$) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in $Γ$. If $Γ$ has global independent offensive alliances, then the \emph{global independent offensive alliance number} $γ_i(Γ)$ is the minimum cardinality among all independent global offensive alliances of $Γ$. In this paper we study mathematical properties of the global (strong) alliance number of cubic graphs. For instance, we show that for all connected cubic graph of order $n$, $$\frac{2n}{5}\le γ_i(Γ)\le \frac{n}{2}\le γ_{\hat{o}}(Γ)\le \frac{3n}{4} \le γ_{\hat{o}}({\cal L}(Γ))=γ_{o}({\cal L}(Γ))\le n,$$ where ${\cal L}(Γ)$ denotes the line graph of $Γ$. All the above bounds are tight.
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