On Eternal Connected Vertex Cover

Abstract

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.