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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.

By using the structural density function (Fraser 1979, Ch. 7) of the parameters of a Pareto distribution, the structural distribution function of the strong Pareto law is derived. Its fractiles have been evaluated numerically for special cases, and the results are displayed through graphs from which structural one‐sided probability bounds may be found. It is shown that these graphs may also be used to find structural tolerance bounds for the Pareto distribution.