AI 中文总结
研究哈林图上的永恒顶点覆盖问题，通过构造一族哈林图证明下界\(\frac{7}{6}\)，给出两种算法证明上界\(\frac{3}{2}\)，其中一种算法对毛毛虫哈林图达到\(\frac{4}{3}\)上界，计算哈林图永恒顶点覆盖数是否NP - 难仍是开放问题。
AI 中文摘要
永恒顶点覆盖问题是一种图保护问题，是经典顶点覆盖问题的动态双人游戏变体。在该游戏中，保护图\(G\)所需的最少警卫数量称为\(G\)的永恒顶点覆盖数，记为\(evc(G)\)。已知对于任何图\(G\)，\(mvc(G) \leq evc(G) \leq 2mvc(G)\)，其中\(mvc(G)\)是\(G\)的顶点覆盖数，且这些界限通常是紧的。然而，没有双连通图\(G\)能达到\(evc(G) = 2mvc(G)\)，且对它们没有更好的下界。本文聚焦于图族中的双连通图，对于无限图族\(\mathcal{F}\)，考虑参数\(\rho(\mathcal{F})=\sup\{r \in \mathbb{R}:\text{ 对于无限多个图 }G \in \mathcal{F},\frac{evc(G)}{mvc(G)}\geq r\}\)。目前还没有已知的双连通图类\(\mathcal{F}\)使得\(1 < \rho(\mathcal{F})<2\)。本文表明当\(\mathcal{F}\)是哈林图族时，\(\frac{7}{6} \leq \rho(\mathcal{F}) \leq \frac{3}{2}\)。哈林图是\(3 -\)连通的且树宽为三。为证明下界，构造了一族哈林图，其比例随图大小增加趋于\(\frac{7}{6}\)。为证明上界，给出了两种算法。第一种算法给出了一种用\(\frac{3}{2} mvc(G)\)个警卫的防御策略，是计算哈林图永恒顶点覆盖数的\(\frac{3}{2}\)因子近似算法，还给出了哈林图几个子类的\(\rho\)的上界\(\frac{4}{3}\)。第二种算法对毛毛虫哈林图达到了\(\frac{4}{3}\)的上界。计算哈林图的永恒顶点覆盖数是否为NP - 难仍是一个开放问题，树宽为二的图也是如此。
英文摘要
Eternal vertex cover problem is a graph protection problem which is a dynamic two player game variant of the classical vertex cover problem. In this game, the minimum number of guards required to protect a graph $G$ is called the eternal vertex cover number of $G$, denoted by $evc(G)$. It is known that for any graph $G$, $\ mvc(G) \le evc(G) \le 2mvc(G)$, where $mvc(G)$ is the vertex cover number of $G$, and that these bounds are generally tight. However, no biconnected graph $G$ achieves $evc(G) = 2mvc(G)$ and no better lower bounds are known for them. In this work, we focus on biconnected graphs in graph families. For infinite graph families $\mathcal{F}$, consider the parameter $ρ(\mathcal{F})=\sup\{r \in \mathbb{R}:\text{ for infinitely many graphs }G \in \mathcal{F},\frac{evc(G)}{mvc(G)}\ge r\}$. No class of biconnected graphs $\mathcal{F}$ is known yet, for which $1 < ρ(\mathcal{F})<2$. In this paper, we show that when $\mathcal{F}$ is the family of Halin graphs, $\frac{7}{6} \le ρ(\mathcal{F}) \le \frac{3}{2}$. Halin graphs are $3$-connected and they have treewidth three. To show the lower bound, we construct a family of Halin graphs for which the ratio tends to $\frac{7}{6}$ with increasing graph size. For the upper bound, we give two algorithms. Our first algorithm gives a defense strategy with $\frac{3}{2} mvc(G)$ guards and serves as a $\frac{3}{2}$ factor approximation algorithm to compute the eternal vertex cover number of Halin graphs. This algorithm also gives an upper bound of $\frac{4}{3}$ for $ρ$ for several subclasses of Halin graphs. Our second algorithm attains the upper bound of $\frac{4}{3}$ for caterpillar Halin graphs. Whether computing eternal vertex cover number is NP-hard for Halin graphs remains an open problem, as is the case with treewidth two graphs.
