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Keywords:

  • the Firefighter Problem;
  • surviving rate;
  • planar graphs

Abstract

In a graph G, a fire starts at some vertex. At every time step, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. The k-surviving rate inline image of G is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs inline image, we are interested in the minimum value k such that for some constant inline image and all inline image, inline image (i.e., such that linearly many vertices are expected to be saved in every graph from inline image). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.