醉醺醺的警察和醉酒的强盗：警察和强盗游戏的变种
Tipsy cop and drunken robber: a variant of the cop and robber game on graphs
摘要：
在YouTube视频\url{ https://www.youtube.com/watch?v=Z_mXDvZQ6dU}中，一个中性粒细胞追逐一个随机方向移动的细菌细胞，受此生物学场景的启发，我们在图上展示了一个警察和强盗游戏的变体，称为醉酒警察和醉酒强盗游戏。在这个游戏中，我们把一个喝醉的警察和一个喝醉的强盗放在一个有限连通图$G$的不同顶点上。游戏由独立的动作组成，强盗从他开始的地方移动到相邻的顶点开始游戏，然后警察从她开始的地方移动到相邻的顶点。由于抢劫犯喝醉了，他在图上进行随机游动，而警察喝醉了意味着她的动作有时是随机的，有时是故意的。我们的主要结果给出了在完全图、完全二部图和圈图等高度对称图上，这个博弈经过$m$移动后，抢劫者仍然没有被抓获的概率公式.我们还给出了这些家庭的图之间的警察和强盗的预期相遇时间。最后，我们提出了一种计算这种概率的一般方法，并详细介绍了未来研究的各种方向。
英文摘要：
Motivated by a biological scenario illustrated in the YouTube video \url{ https://www.youtube.com/watch?v=Z_mXDvZQ6dU} where a neutrophil chases a bacteria cell moving in random directions, we present a variant of the cop and robber game on graphs called the tipsy cop and drunken robber game. In this game, we place a tipsy cop and a drunken robber at different vertices of a finite connected graph $G$. The game consists of independent moves where the robber begins the game by moving to an adjacent vertex from where he began, this is then followed by the cop moving to an adjacent vertex from where she began. Since the robber is inebriated, he takes random walks on the graph, while the cop being tipsy means that her movements are sometimes random and sometimes intentional. Our main results give formulas for the probability that the robber is still free from capture after $m$ moves of this game on highly symmetric graphs, such as the complete graphs, complete bipartite graphs, and cycle graphs. We also give the expected encounter time between the cop and robber for these families of graphs. We end the manuscript by presenting a general method for computing such probabilities and also detail a variety of directions for future research.
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