The game of Overprescribed Cops and Robbers played on graphs

TL;DR

This paper investigates how increasing the number of cops affects the duration of the Cops and Robbers game on various graph classes, providing exact and asymptotic results for capture times.

Contribution

It characterizes capture times for trees, grids, hypercubes, and random graphs with many cops, extending understanding of pursuit-evasion dynamics on different graph structures.

Findings

Capture time decreases monotonically as more cops are added.

Full characterization of capture times for trees with any number of cops.

Asymptotic capture times for grids, hypercubes, and binomial random graphs.

Abstract

We consider the effect on the length of the game of Cops and Robbers when more cops are added to the game play. In Overprescribed Cops and Robbers, as more cops are added, the capture time (the minimum length of the game assuming optimal play) monotonically decreases. We give the full range of capture times for any number of cops on trees, and classify the capture time for an asymptotic number of cops on grids, hypercubes, and binomial random graphs. The capture time of planar graphs with a number of cops at and far above the cop number is considered.