Cops and Robbers on 1-Planar Graphs

Stephane Durocher, Shahin Kamali, Myroslav Kryven, Fengyi Liu, Amirhossein Mashghdoust, Avery Miller, Pouria Zamani Nezhad, Ikaro Penha Costa, Timothy Zapp

Abstract: 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 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 completely determine the cop number of outer 1-planar graphs.