Lions and contamination, triangular grids, and Cheeger constants

TL;DR

This paper investigates the minimum number of lions needed to clear contamination from a graph, establishing a lower bound based on the Cheeger constant and extending previous grid graph analyses to triangular grids.

Contribution

It introduces a Cheeger constant-based lower bound for lion requirements and extends contamination clearing analysis from square to triangular grid graphs.

Findings

Lower bound on lions needed in terms of Cheeger constant

Extension of analysis to triangular grid graphs

Generalization of previous square grid results

Abstract

Suppose each vertex of a graph is originally occupied by contamination, except for those vertices occupied by lions. As the lions wander on the graph, they clear the contamination from each vertex they visit. However, the contamination simultaneously spreads to any adjacent vertex not occupied by a lion. How many lions are required in order to clear the graph of contamination? We give a lower bound on the number of lions needed in terms of the Cheeger constant of the graph. Furthermore, the lion and contamination problem has been studied in detail on square grid graphs by Brass et al. and Berger et al., and we extend this analysis to the setting of triangular grid graphs.