The 2-surviving rate of planar graphs without 4-cycles

Let G be a connected graph with n≥2 vertices. 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 two 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 sn2(v) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The surviving rate ρ2(G) of G is defined to be , which is the average proportion of saved vertices.In this paper, we show that if G is a planar graph with n≥2 vertices and without 4-cycles, then .