In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.

翻译：在警察与强盗游戏中，最有用的结论之一是图中的一个等距路径可由一名警察守护。本文引入了子图中宽影的概念，并利用该概念刻画了所有可单警守护的图。作为应用，我们证明了在任意平面图中，三名警察即可捕获强盗，且附加限制为最多允许两名警察同时移动，这证实了杨的猜想并强化了艾格纳与弗罗姆的经典结果。