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. This result also holds for split graphs. 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种动态变体。我们考虑一个双人博弈：守卫被放置在图中的若干顶点上。每一步中，一名玩家（攻击者）攻击一条边。作为回应，另一名玩家（防御者）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。若无法进行这样的移动，则攻击者获胜；若防御者能够防守住无限序列的攻击，则防御者获胜。防御者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。在一般图上，确定最小永恒顶点覆盖数的计算问题是NP难问题，并存在一个2-近似算法和一个指数级核。该问题在二分图上的复杂性以及是否承认多项式核的问题尚未解决。我们通过证明永恒顶点覆盖问题在直径为六的二分图上也是NP难且不承认多项式压缩，解决了这两个问题。这一结果同样适用于分裂图。此外，我们还证明该问题在余二分图类上存在多项式时间算法。