The localization game is a variant of the game of Cops and Robber in which the robber is invisible and moves between adjacent vertices, but the cops can probe any $k$ vertices of the graph to obtain the distance between probed vertices and the robber. The localization number of a graph is the minimum $k$ needed for cops to be able to locate the robber in finite time. The localization capture time is the number of rounds needed for cops to win. The localization capture time conjecture claims that there exists a constant $C$ such that the localization number of every connected graph on $n$ vertices is at most $Cn$. While it is known that the conjecture holds for trees, in this paper we significantly improve the known upper bound for the localization capture time of trees. We also prove the conjecture for a subclass of outerplanar graphs and present a generalization of the localization game that appears useful for making further progress towards the conjecture.

翻译：定位游戏是“警察与强盗”游戏的一种变体，其中强盗不可见且只能在相邻顶点间移动，但警察可以探测图中任意 $k$ 个顶点，以获取被探测顶点与强盗之间的距离。图的定位数是指警察能在有限时间内定位强盗所需的最小 $k$ 值。定位捕获时间则是指警察获胜所需的回合数。定位捕获时间猜想断言：存在一个常数 $C$，使得每个具有 $n$ 个顶点的连通图的定位数至多为 $Cn$。尽管已知该猜想对树成立，本文中我们显著改进了树定位捕获时间的已知上界。我们还证明了该猜想对一类外部平面图子图成立，并提出了一种定位游戏的推广形式，该形式似乎有助于进一步推进该猜想的证明。