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. A second result is that in the class of graphs with bounded degree, having the constant containment property is closed under quasi-isometry. We prove analogous results for the {fn}-containment property, where fn 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 Ld does not satisfy the cnd−3-containment property for any constant c.