We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph $G$ and is denoted by $evc(G)$. It is clear that $evc(G)$ is at least $mvc(G)$, the size of a minimum vertex cover of $G$. We say that $G$ is Spartan if $evc(G) = mvc(G)$. The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on $2n$ vertices where every edge belongs to a perfect matching, an easy strategy is to have $n$ guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.

翻译：我们研究图上一类攻击者与防御者之间的双人博弈。初始时，防御者在顶点子集上放置守卫。每一步中，攻击者攻击一条边。防御者必须至少移动一个守卫穿过被攻击的边以防御攻击。当且仅当防御者能够防御无限序列的攻击时，防御者获胜。防御者拥有获胜策略所需的最少守卫数量称为图$G$的永恒顶点覆盖数，记为$evc(G)$。显然，$evc(G)$至少为$mvc(G)$，即$G$的最小顶点覆盖的大小。若$evc(G) = mvc(G)$，则称$G$为Spartan图。Spartan图的刻画问题在很大程度上仍未解决。对于$2n$个顶点上每条边均属于一个完美匹配的二部图，一个简单的策略是设置$n$个守卫，这些守卫在应对攻击时始终沿着完美匹配移动。我们证明这些本质上就是唯一的Spartan二部图。