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

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 $c$.