图上的僵尸和幸存者游戏的概率版本
A probabilistic version of the game of Zombies and Survivors on graphs
摘要：
我们考虑一个新的概率图搜索游戏的图形，灵感来自熟悉的游戏警察和强盗。在Zombies and Survivors中，一组僵尸试图在给定的图形上吃掉一个孤独的幸存者。僵尸随机选择他们的初始位置，并在游戏过程中，直接向幸存者移动。在每一轮，他们移动到邻近的顶点，最小化到幸存者的距离;如果有一个以上的这样的顶点，那么他们随机选择一个。幸存者试图通过移动到邻近的顶点或留在他当前的顶点来逃离僵尸。如果最终有一个僵尸落在它们的顶点上吃掉了幸存者，僵尸就赢了;否则，幸存者就赢了。图的僵尸数是游戏所需的僵尸的最小数量，使得它们获胜的概率严格大于1/2。我们提出的僵尸数的几个图形的家庭，如周期，超立方体，投影平面的关联图，笛卡尔和环形网格的渐近结果。
英文摘要：
We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is strictly greater than 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
