Eternal Vertex Cover problem is a dynamic variant of the vertex cover problem. We have a two player game in which guards are placed on some vertices of a graph. In every move, one player (the attacker) attacks an edge. In response to the attack, the second player (defender) moves the guards along the edges of the graph in such a manner that at least one guard moves along the attacked edge. If such a movement is not possible, then the attacker wins. If the defender can defend the graph against an infinite sequence of attacks, then the defender wins. The minimum number of guards with which the defender has a winning strategy is called the Eternal Vertex Cover Number of the graph G. On general graphs, the computational problem of determining the minimum eternal vertex cover number is NP-hard and admits a 2-approximation algorithm and an exponential kernel. The complexity of the problem on bipartite graphs is open, as is the question of whether the problem admits a polynomial kernel. We settle both these questions by showing that Eternal Vertex Cover is NP-hard and does not admit a polynomial compression even on bipartite graphs of diameter six. We also show that the problem admits a polynomial time algorithm on the class of cobipartite graphs.

翻译：永恒顶点覆盖问题是顶点覆盖问题的一个动态变体。我们考虑一个双人博弈：在图的某些顶点上放置守卫。每轮攻击方攻击一条边，防御方需沿图的边移动守卫，使得至少有一个守卫沿被攻击的边移动。若无法实现该移动，则攻击方获胜；若防御方可抵御无限次攻击序列，则防御方获胜。防御方拥有获胜策略所需的最少守卫数称为图G的永恒顶点覆盖数。对于一般图，确定最小永恒顶点覆盖数的计算问题是NP难的，且存在2-近似算法和指数级核。该问题在二分图上的复杂度尚未明确，其是否允许多项式核也悬而未决。我们通过证明即使对直径为6的二分图，永恒顶点覆盖问题也是NP难且不存在多项式压缩，从而解决了这两个问题。同时表明该问题在共二分图类上存在多项式时间算法。