将图划分为防御性k联盟
Partitioning a graph into defensive k-alliances
摘要：
图中的防御性$k$ -联盟是一个顶点集合$S$，它具有这样的性质：$S$中的每个顶点在$S$中的邻居至少比它在$S$之外的邻居多$k$。一个防御性的$k$ -联盟$S$称为全局的，如果它形成一个支配集。在本文中，我们研究的问题划分的顶点集的图（全球）防御$k$ -联盟。图$Γ=(V,E)$的（全局）防御$k$ -联盟划分数（$ψ_{k}^{gd}(Γ)$）$ψ_k^{d}(Γ)$定义为$V$的一个划分中使得每个集合是（全局）防御$k$ -联盟的最大集合数.我们得到了$ψ_k^{d}(Γ)$和$ψ_{k}^{gd}(Γ)$关于图的阶数、大小、最大度、最小度、代数连通度和等周数的紧界.此外，我们还研究了存在于$Γ_1\times Γ_2$划分成（全球）防御性$(k_1+k_2)$ -联盟和$Γ_i$划分成（全球）防御性$k_i$ -联盟，$i\in \{1,2\}$之间的密切关系。
英文摘要：
A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k$-alliances. The (global) defensive $k$-alliance partition number of a graph $Γ=(V,E)$, ($ψ_{k}^{gd}(Γ)$) $ψ_k^{d}(Γ)$, is defined to be the maximum number of sets in a partition of $V$ such that each set is a (global) defensive $k$-alliance. We obtain tight bounds on $ψ_k^{d}(Γ)$ and $ψ_{k}^{gd}(Γ)$ in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of $Γ_1\times Γ_2$ into (global) defensive $(k_1+k_2)$-alliances and partitions of $Γ_i$ into (global) defensive $k_i$-alliances, $i\in \{1,2\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
