Cops and Robbers is a well-studied pursuit-evasion game in which a set of cops seeks to catch a robber in a graph G, where cops and robber move along edges of G. The cop number of G is the minimum number of cops that is sufficient to catch the robber. Every planar graph has cop number at most three, and there are planar graphs for which three cops are necessary [Aigner and Fromme, DAM 1984]. We study the problem for beyond-planar graphs, that is, graphs that can be drawn in the plane with few crossings. In particular, we focus on 1-planar graphs, that is, graphs that can be drawn in the plane with at most one crossing per edge. In contrast to planar graphs, we show that some 1-planar graphs have unbounded cop number. Meanwhile, for maximal 1-planar graphs, we prove that three cops are always sufficient and sometimes necessary. In addition, we characterize outer 1-planar graphs with respect to their cop number.

翻译：警察与强盗是一个被广泛研究的追逃博弈问题，其中一组警察试图在图 G 中抓获一名强盗，警察和强盗均沿 G 的边移动。图 G 的警察数是指足以抓获强盗所需的最少警察人数。每个平面图的警察数至多为3，且存在一些平面图需要3名警察才能抓获[Aigner and Fromme, DAM 1984]。我们研究了超平面图（即可以在平面上绘制且仅有少量交叉的图）中的这个问题。具体而言，我们聚焦于1-平面图，即每条边在平面上至多有一个交叉的图。与平面图不同，我们发现某些1-平面图具有无界的警察数。同时，对于最大1-平面图，我们证明3名警察总是足够的，有时也是必要的。此外，我们根据警察数对外部1-平面图进行了刻画。