Cops that surround a robber

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 de...

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Bibliographic Details
Published in:arXiv.org 2021-08
Main Authors: Burgess, Andrea C, Cameron, Rosalind A, Clarke, Nancy E, Danziger, Peter, Finbow, Stephen, Jones, Caleb W, Pike, David A
Format: Article
Language:English
Subjects:
Apexes
Combinatorial analysis
Games
Graphs
Robbery
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1910.14200