The 2‐surviving rate of planar graphs with average degree lower than
Przemysław Gordinowicz · Journal of Graph Theory · 2018
Abstract Let G be any connected graph on n vertices, . Let k be any positive integer. Suppose that a fire breaks out at some vertex of G . Then, in each turn firefighters can protect at most k vertices of G not yet on fire; Next the fire spreads to all unprotected neighbors of burning vertices. The k ‐surviving rate of G, denoted by , is the expected fraction of vertices that can be saved from the fire, provided that the starting vertex is chosen uniformly at random. In this note, it is shown that for any planar graph G with average degree , where , there is . In particular, the result implies a significant improvement of the bound for 2‐surviving rate for triangle‐free planar graphs (Esperet et al. ) and for planar graphs without 4‐cycles (Kong et al. ). The proof is done using the separator theorem for planar graphs.