Cops and Robbers on 1-Planar Graphs

TL;DR

This paper investigates the cops and robbers pursuit game on 1-planar graphs, revealing that some have unbounded cop number while maximal 1-planar graphs require at most three cops, similar to planar graphs.

Contribution

It establishes bounds on the cop number for 1-planar graphs, including maximal and outer 1-planar cases, extending understanding beyond planar graphs.

Findings

Some 1-planar graphs have unbounded cop number.

Maximal 1-planar graphs require at most three cops.

Outer 1-planar graphs are characterized by their cop number.

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 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.