The Surviving Rate of a Graph for the Firefighter Problem

We consider the following firefighter problem on a graph $G=(V,E)$. Initially, a fire breaks out at a vertex v of G. In each subsequent time unit, a firefighter protects one vertex, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. Let $\mathrm{sn}(v)$ denote the maximum number of vertices the firefighter can save when a fire breaks out at vertex v of G. We define the surviving rate $\rho(G)$ of G to be the average percentage of vertices that can be saved when a fire randomly breaks out at a vertex of G, i.e., $\rho(G)=\sum_{v\in V}\mathrm{sn}(v)/n^2$. In this paper, we prove that for every tree T on n vertices, $\rho(T)>1-\sqrt{2/n}$. Furthermore, we show that $\rho(G)>1/6$ for every outerplanar graph G, and $\rho(H)>3/10$ for every Halin graph H with at least 5 vertices.