Cops that surround a robber

TL;DR

This paper introduces the Surrounding Cops and Robbers game on graphs, where cops aim to occupy all neighbors of a robber, and studies the surrounding cop number across various graph classes.

Contribution

It defines the surrounding cop number and provides bounds and exact values for different graph classes, expanding understanding of pursuit-evasion dynamics.

Findings

Bounds on the surrounding cop number for general graphs

Exact values for specific graph classes such as product graphs and Petersen graphs

Insights into how graph structure influences pursuit strategies

Abstract

We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber's neighbouring vertices. We denote by $\sigma(G)$ the {\em surrounding cop number} of $G$, namely the least number of cops required to surround a robber in the graph $G$. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest include product graphs, graphs arising from combinatorial designs, and generalised Petersen graphs.