Firefighting on a random geometric graph

Let G λ be the graph whose vertices are points of a planar Poisson process of density λ , with vertices adjacent if they are within distance 1. A “fire” begins at some vertex and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time‐step. Let f λ be the least f such that, with probability 1, any fire on G λ can be stopped in finite time. We show that f λ is bounded between two linear functions of λ . The lower bound makes use of a new result concerning oriented percolation in the plane; the constant factor in the upper bound is is tight, provided a certain conjecture, for which we offer supporting evidence, is correct. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 466–477, 2015