Plane graphs of diameter two are 2-optimal

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Let G be a connected graph with n vertices. Suppose a fire breaks out at some vertex of G. At each time interval, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. Let dnk(v) denote the minimum number of vertices that the fire may damage when a fire breaks out at vertex v. The k-expected damage of G, denoted by εk(G), is the expectation of the proportion of vertices that can be damaged from the fire, if the starting vertex of the fire is chosen uniformly at random, i.e., εk(G)=∑v∈V(G)dnk(v)/n2. A class of graphs G is called k-optimal if εk(G) tends to 0 as n tends to infinity for any G∈G. In this paper, we prove that planar graphs of diameter two are 2-optimal, which is the best possible.