The Coarse Geometry of Hartnell's Firefighter Problem on Infinite Graphs

TL;DR

This paper investigates the fire containment problem on infinite graphs using geometric and group theoretic concepts, establishing conditions under which graphs can contain fires and how these properties are preserved under quasi-isometry.

Contribution

It introduces the concept of constant containment property for graphs, characterizes it for graphs with quadratic growth, and shows its invariance under quasi-isometry, extending previous results.

Findings

Quadratic growth graphs have the constant containment property.

Quasi-isometry preserves the containment property in bounded degree graphs.

Regular Euclidean tilings in 2D have the containment property, higher dimensions do not.

Abstract

In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the $n$-containment property if for every finite initial fire, there is a strategy to contain the fire by protecting $n$ vertices at each turn. A graph has the constant containment property if there is an integer $n$ such that it has the $n$-containment property. Our first result is that any locally finite connected graph with quadratic growth has the constant containment property; the converse does not hold. This result provides a unified way to recover previous results in the literature, in particular the class of graphs satisfying the constant containment property is infinite. A second result is that in the class of graphs with bounded degree, having the constant containment property is preserved by quasi-isometry. Some sample consequences of the second result are that any regular tiling of the Euclidean plane has the fire containment property; no regular tiling of the $n$-dimensional Euclidean space has the containment property if $n>2$; and no regular tiling of the $n$-dimensional hyperbolic space has the containment property if $n\geq 2$. We prove analogous results for the $\{f_n\}$-containment property, where $f_n$ is an integer sequence corresponding to the number of vertices protected at time $n$. In particular, we positively answer a conjecture by Develin and Hartke by proving that the $d$-dimensional square grid $\mathbb{L}^d$ does not satisfy the $cn^{d-3}$-containment property for any constant $c$.