Fire containment in grids of dimension three and higher

TL;DR

This paper investigates fire containment strategies in high-dimensional grids, proving several conjectures and analyzing the minimal burnt vertices when deploying limited firefighters in a deterministic model.

Contribution

It extends fire containment analysis to three or more dimensions, proving conjectures and providing new insights into optimal firefighting strategies in high-dimensional lattices.

Findings

Proved two conjectures of Wang and Moeller [2002].

Analyzed minimal burnt vertices with limited firefighters.

Extended fire spread models to higher dimensions.

Abstract

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [1995] and the problem of minimizing the number of burnt vertices when deploying a limited number of firefighters per timestep. We consider the process occurring on the d-dimensional square lattice for d>=3, and we prove several results, including two conjectures of Wang and Moeller [2002].