论文标题
在大地间空间上的警察和强盗游戏
The game of Cops and Robber on geodesic spaces
论文作者
论文摘要
传统上，警察和强盗的游戏是在有限图上进行的。本文的目的是介绍和分析在任意的大地测量空间（一个紧凑的，路径连接的空间赋予内在度量标准）上进行的游戏。结果表明，在公制图上玩的游戏基本上与在抽象图上玩的离散游戏相同，并且对于每个紧凑的地球表面都有一个整数$ c $，因此$ c $ cops可以赢得一场强盗的胜利，而$ c $仅取决于表面的$ g $。结果表明，$ c = 3 $用于属$ 0 $或$ 1 $的可定向表面以及crosscap数字的不可方向的表面$ 1 $或$ 2 $（带有任何数量的边界组件），以及$ c = o（g）$和$ c =ω（\ sqrt {g} $ n时$ g $ g $ g $更大。讨论此游戏的主要动机是查看COP号码（捕获强盗所需的最小警察数量）是一个新的几何不变式，描述了地球空间的复杂程度。
The game of Cops and Robber is traditionally played on a finite graph. The purpose of this paper is to introduce and analyse the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer $c$ such that $c$ cops can win the game against one robber, and $c$ only depends on the genus $g$ of the surface. It is shown that $c=3$ for orientable surfaces of genus $0$ or $1$ and nonorientable surfaces of crosscap number $1$ or $2$ (with any number of boundary components) and that $c=O(g)$ and that $c=Ω(\sqrt{g})$ when the genus $g$ is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space.
