Eternally surrounding a robber

TL;DR

This paper introduces the bodyguard problem on graphs, focusing on surrounding and indefinitely protecting a target, and provides solutions for various graph families and product graphs.

Contribution

It formulates the bodyguard problem, analyzes bounds, and solves it for multiple graph classes and graph product types, extending the surrounding cops and robber framework.

Findings

Solved for complete graphs, wheels, trees, cycles, and multipartite graphs.

Analyzed bounds and strategies for indefinite surrounding.

Extended the problem to Cartesian, strong, and lexicographic products.

Abstract

We introduce the bodyguard problem for graphs. This is a variation of Surrounding Cops and Robber but, in this model, a smallest possible group of bodyguards must surround the president and then maintain this protection indefinitely. We investigate some elementary bounds, then solve this problem for the infinite graph families of complete graphs, wheels, trees, cycles, complete multipartite graphs, and two-dimensional grids. We also examine the problem in more general Cartesian, strong, and lexicographic products.