**Abstract:** |
Let G be a connected graph with vertices. Let 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 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 of G is defined to be , 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, d-degenerate graphs, general planar graphs, etc. |