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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 -containment
property if for every finite initial fire, there is a strategy to contain the
fire by protecting vertices at each turn. A graph has the constant
containment property if there is an integer such that it has the
-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 -dimensional Euclidean space has the
containment property if ; and no regular tiling of the -dimensional
hyperbolic space has the containment property if . We prove analogous
results for the -containment property, where is an integer
sequence corresponding to the number of vertices protected at time . In
particular, we positively answer a conjecture by Develin and Hartke by proving
that the -dimensional square grid does not satisfy the
-containment property for any constant .Comment: Version accepted by Discrete Mathematic