Accelerated Cops and Robbers

TL;DR

This paper introduces a variant of the Cops and Robbers game where both players can move multiple edges per turn, providing new bounds and results for specific graph classes like grids and hypercubes.

Contribution

It extends the classic Cops and Robbers model to allow multiple-edge moves and establishes bounds for cop numbers and capture times in this new setting.

Findings

Derived bounds for cop numbers on grids and hypercubes.

Determined capture time for cop-win graphs when s=2.

Provided general results applicable to the new multi-edge move model.

Abstract

We consider a variant of Cops and Robbers in which both the cops and the robber are allowed to traverse up to $s$ edges on each of their turns, where $s \ge 2$. We give several general for this new model as well as establish bounds for the cop numbers for grids and hypercubes. We also determine the capture time of cop-win graphs when $s = 2$ up to a small additive constant.