The coarse geometry of Hartnell’s firefighter problem on infinite graphs

Abstract

Abstract. In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the ncontainment 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 L d does not satisfy the cn d−3 -containment property for any constant c.

Fire retainment on Cayley graphs

“…The graph G is said to satisfy the polynomial containment of degree d (respectively, polynomial retainment of degree d) if it has the {f (n)}containment property (respectively, {f (n)}-retainment property) for f (n) = Kn d , for some K ≥ 0. Dyer, Martínez-Pedroza, and Thorne [3] established that having polynomial containment of degree d is a quasi-isometry invariant, while the analogous statement for polynomial retainment of degree d was proved in [8].…”

The firefighter problem on polynomial and intermediate growth groups

“…Partial results were proved by Dyer, Martinez-Pedroza and Thorne in [3]. To that end they studied a spherical version of isoperimetry, looking at the number of neighbours of subsets of S n inside S n+1 .…”

“…Note that by results of [3] containment is a quasi-isometry invariant, in the following sense: define g f if there exists C > 0 s.t. g(x) ≤ Cf (Cx) and g ≃ f if g f and f g. Theorem 4.4. of [3] states that if G 1 is quasiisometric to G 2 , and G 1 satisfies the {f (n)}-containment property, then G 2 satisfies the {g(n)}-containment property for some g ≃ f .…”

A note on the relation between Hartnell’s firefighter problem and growth of groups

“…If the graph G has the containment property for a sequence of the form fn = Cn d , then the graph is said to have polynomial containment. In [5], it is shown that any locally finite graph with polynomial growth has polynomial containment; and it is remarked that the converse does not hold. That article also raised the question of whether the equivalence of polynomial growth and polynomial containment holds for Cayley graphs of finitely generated groups.…”

“…Theorem 1. [5,Theorem 8] In the class of connected graphs with bounded degree, the property of having polynomial containment of degree at most d is preserved by quasi-isometry.…”