Planar graphs without chordal 5-cycles are 2-good

Abstract

Let G be a connected graph with $n\ge 2$ 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$_2(v)$ 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 $\rho _2(G)$ of G is defined to be the real number $\frac{1}{n^2} \sum _{v\in V(G)} \mathrm{sn}_2(v)$. Then it is obvious that $0<\rho _2(G)<1$. The graph G is called 2-good if there is a constant $c>0$ such that $\rho _2(G)>c$. In this paper, we prove that every planar graph with $n\ge 2$ vertices and without chordal 5-cycles is 2-good.