Abstract:
 Let G be a connected graph with $n\geq 2$ vertices. Let $k\ge 1$ 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 kvertices 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 ksurviving rate$\rho_k(G)$ of G is defined to be $\Sum_{v\in V(G)}sn_k(v)/n^2$, which is the average proportion of saved vertices. In this talk, we give a chief survey on this direction and related problems. In particular, we consider the firefighter problems for some special graphs such as trees, outerplanar graphs, planar graphs of large girth, ddegenerate graphs, general planar graphs, etc.
