Surviving rates of trees and outerplanar graphs for the firefighter problem

The firefighter problem is a discrete-time game on graphs introduced by Hartnell in an attempt to model the spread of fire, diseases, computer viruses and suchlike in a macro-control level. To measure the defence ability of a graph as a whole, Cai and Wang defined the surviving rate of a graph G for the firefighter problem to be the average percentage of vertices that can be saved when a fire starts randomly at one vertex of G. In this paper, we prove that the surviving rate of every n-vertex outerplanar graph is at least 1 − Θ( log n n ), which is asymptotically tight. We also show that the greedy strategy of Hartnell and Li for trees saves at least 1 − Θ( log n n ) percentage of vertices on average for an n-vertex tree.