发表机构
Indian Institute of Science, Bengaluru（印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究永恒连通顶点覆盖数，证明每个顶点属于某个最小连通顶点覆盖并非充分条件，并刻画了达到锐界的图类，给出若干图族的等号成立条件。
AI 中文摘要
对于至少有一条边的连通图$G$，永恒连通顶点覆盖数$ecvc(G)$是在对任意边攻击序列的每次响应后，能够维持一个连通顶点覆盖的最少守卫数。记连通顶点覆盖的最小大小为$cvc(G)$。已知$cvc(G)\leq ecvc(G)\leq cvc(G)+1$。$ecvc(G)=cvc(G)$的一个必要条件是每个顶点都属于某个最小连通顶点覆盖。我们证明这个条件不是充分的：存在一个32个顶点的图$G$，其$cvc(G)=19$且$ecvc(G)=20$，尽管$G$的每个顶点都属于某个最小连通顶点覆盖。对于最小度至少为2的连通图，我们建立了锐界$cvc(G)\geq 2|V(G)|-|E(G)|-1$，并用$\mathcal F$表示达到等号的图类。我们证明$G\in\mathcal F$当且仅当度数至少为3的顶点导出一个森林。在$\mathcal F$内，条件$ecvc(G)=cvc(G)$、每个顶点都属于某个最小连通顶点覆盖、以及每个环上至少存在两个度为2的顶点，这三者是等价的。作为应用，我们得到了对于最小度至少为2的连通图的完全细分图以及极小2-连通图，有$ecvc(G)=cvc(G)$。在这两个图族中，每个最小连通顶点覆盖都是一个永恒获胜配置。对于$\mathcal{F}$之外的图，这通常不成立，我们给出了这样一个图的一个例子。
英文摘要
For a connected graph $G$ with at least one edge, the \textit{eternal connected vertex cover} number $ecvc(G)$ is the minimum number of guards that can maintain a connected vertex cover after every response to an arbitrary sequence of edge attacks. Denote the minimum size of a connected vertex cover by $cvc(G)$. It is known that $cvc(G)\leq ecvc(G)\leq cvc(G)+1$. A necessary condition for $ecvc(G)=cvc(G)$ is that every vertex belongs to some minimum connected vertex cover. We show that this condition is not sufficient: there exists a $32$-vertex graph $G$ that has $cvc(G)=19$ and $ecvc(G)=20$, although every vertex of $G$ belongs to some minimum connected vertex cover. For connected graphs with minimum degree at least two, we establish the sharp bound $cvc(G)\geq 2|V(G)|-|E(G)|-1$ and $\mathcal F$ denotes the class attaining equality. We prove that $G\in\mathcal F$ if and only if the vertices of degree at least three induce a forest. Within $\mathcal F$, the conditions $ecvc(G)=cvc(G)$, membership of every vertex in some minimum connected vertex cover, and the presence of at least two degree-two vertices on every cycle are equivalent. As applications, we obtain $ecvc(G)=cvc(G)$ for full subdivisions of connected graphs of minimum degree at least two and for minimally $2$-connected graphs. In both the families, every minimum connected vertex cover is an eternally winning configuration. This is not true in general for graphs outside $\mathcal{F}$, we show one example of such a graph.
