AI 中文总结
本文提出定向定位游戏，研究弦图、笛卡尔积图等图类的定向定位游戏，并基于退化度和树宽界定图的定向定位数。
AI 中文摘要
在图G上的定位游戏中，一队警察通过“探测”顶点来搜寻G上看不见的移动强盗；每次探测会告知警察被探测顶点到强盗的距离，若警察能唯一确定强盗的位置则获胜。本文引入了一个相关游戏：定向定位游戏。在该游戏中，探测返回的不是距离，而是方向：当警察探测顶点v时，强盗必须回应v的一个或多个邻居，这些邻居位于从v到强盗位置的最短路径上。G上赢得该游戏所需的最少警察数量称为G的定向定位数。我们研究了几类图上的定向定位游戏，包括弦图、笛卡尔积图和射影平面的关联图，还基于图G的退化度和树宽对G的定向定位数进行了界定。
英文摘要
In the localization game on a graph $G$, a team of cops searches for an invisible, mobile robber on $G$ by "probing" vertices; each probe tells the cops the distance from the probed vertex to the robber. The cops win if they can uniquely determine the robber's location. In this paper, we introduce a related game: the directional localization game. In this game, instead of probes returning distances, they return directions: when the cops probe a vertex $v$, the robber must respond with one or more neighbors of $v$ that lie on a shortest path from $v$ to the robber's location. The minimum number of cops needed to win this game on $G$ is the directional localization number of $G$. We study the directional localization game on several classes of graphs, including chordal graphs, Cartesian products, and incidence graphs of projective planes. We also bound the directional localization number of a graph $G$ in terms of the degeneracy and the treewidth of $G$.
