Cops and Robbers on Intersection Graphs

TL;DR

This paper studies the maximum number of cops needed to catch a robber on various classes of geometric intersection graphs, establishing bounds for some classes and unboundedness for others.

Contribution

It provides new bounds on cop numbers for classes of geometric intersection graphs and demonstrates unbounded cop numbers in certain higher-dimensional cases.

Findings

String graphs have cop number at most 15.

Outer-string graphs have cop number between 3 and 4.

Intersection graphs of 3D objects can have unbounded cop number.

Abstract

The cop number of a graph $G$ is the smallest $k$ such that $k$ cops win the game of cops and robber on $G$. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most $10g+15$ in case of orientable surface of genus $g$, and at most $10g'+15$ in case of non-orientable surface of Euler genus $g'$. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.