Planar graphs without chordal 5-cycles are 2-good

Let G be a connected graph with vertices. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each step, the firefighter protects two vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let sn denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The 2-surviving rate of G is defined to be the real number . Then it is obvious that . The graph G is called 2-good if there is a constant such that . In this paper, we prove that every planar graph with vertices and without chordal 5-cycles is 2-good.