The surviving rate of an outerplanar graph for the firefighter problem

Let G be a connected graph with n>=2 vertices. Let k>=1 be an integer. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each time interval, the firefighter protects k-vertices not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let sn"k(v) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The k-surviving rate @r"k(G) of G is defined to be @?"v"@?"V"("G")sn"k(v)/n^2, which is the average proportion of saved vertices. In this paper, we investigate the surviving rate of outerplanar graphs G with n>=2 vertices. The main results are as follows: (1) lim"n"->"[email protected]"5(G)=1; and (2) @r"1(G)>=4381-53n+3n^2 if n>=8, and @r"1(G)>=13 if n>=2, which improves the result in [L.Cai, W.Wang, The surviving rate of a graph for the firefighter problem, SIAM J. Discrete Math. 23 (2009) 1814-1826].