关于仙人掌图的m-永恒控制数
On the m-eternal Domination Number of Cactus Graphs
摘要：
给定一个图$G$，保护被放置在$G$的顶点上。然后顶点受到无限序列的攻击，使得每个攻击必须由从相邻顶点移动的守卫来防御。m-永恒控制数是使图可以被无限防御的最小守卫数。在本文中，我们研究仙人掌图的m-永恒控制数，即连通图的每条边位于最多两个圈，我们考虑了三个变种的m-永恒控制数：第一个变种允许多个警卫占领一个单一的顶点，第二个变种不允许它，并在第三个变种额外的“驱逐”攻击必须防御。我们给出了仙人掌图的m-永恒控制数的一个新的上界，并证明了对于一个叫做Christmas仙人掌图的仙人掌图的子类，其中每个顶点至多位于两个圈内，这三个数是相等的.此外，我们提出了一个线性时间算法来计算它们。
英文摘要：
Given a graph $G$, guards are placed on vertices of $G$. Then vertices are subject to an infinite sequence of attacks so that each attack must be defended by a guard moving from a neighboring vertex. The m-eternal domination number is the minimum number of guards such that the graph can be defended indefinitely. In this paper we study the m-eternal domination number of cactus graphs, that is, connected graphs where each edge lies in at most two cycles, and we consider three variants of the m-eternal domination number: first variant allows multiple guards to occupy a single vertex, second variant does not allow it, and in the third variant additional "eviction" attacks must be defended. We provide a new upper bound for the m-eternal domination number of cactus graphs, and for a subclass of cactus graphs called Christmas cactus graphs, where each vertex lies in at most two cycles, we prove that these three numbers are equal. Moreover, we present a linear-time algorithm for computing them.
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