The surviving rate of planar graphs

The following firefighter problem on a finite graph G = (V,E) was introduced by Hartnell at the conference in 1995 [3]. Suppose that a fire breaks out at a given vertex v ∈ V . In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then fire spreads to all unprotected neighbours of the vertices on fire. (Once a vertex is on fire or gets protected it stays in such state forever.) Since the graph is finite, at some point each vertex is either on fire or is protected by the firefighter, and the process is finished. (Alternatively, one can stop the process when no neighbour of the vertices on fire is unprotected. The fire will no longer spread.) The objective of the firefighter is to save as many vertices as possible. Today, 15 years later, our knowledge about this problem is much greater and a number of papers have been published. We would like to refer the reader to the survey of Finbow and MacGillivray for more information [6]. We would like to focus on the following property. Let sn(G, v) denote the number of vertices in G the firefighter can save when a fire breaks out at vertex v ∈ V , assuming the best strategy is used. The surviving rate ρ(G) of G, introduced in [5], is defined as the expected percentage of vertices that can be saved when a fire breaks out at a random vertex of G (uniform distribution is used), that is, ρ(G) = 1 n2 ∑ v∈V sn(G, v). It is not difficult to see that for cliques ρ(Kn) = 1 n , since no matter where a fire breaks out only one vertex can be saved. For paths we get that