Firefighting on Geometric Graphs with Density Bounds

Let G be an infinite geometric graph; in particular, a graph whose vertices are a countable discrete set of points on the plane, with vertices u, v adjacent if their Euclidean distance is less than 1. A “fire” begins at some finite set of vertices and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time-step. Let f(G) be the least f for which any fire on G can be stopped in finite time. We show that if G has bounded density, in the sense that no open disk of radius r contains more than λ vertices, then f(G) is bounded above by the ceiling of a universal constant times λ/r. Similarly, if the density of G is bounded from below in the sense that every open disk of radius r contains at least κ vertices, then f(G) is bounded below by κ times the square of the floor of a universal constant times 1/r.