The surviving rate of digraphs

Abstract

Let G⃗ be a connected digraph with n≥2 vertices. Suppose that a fire breaks out at a vertex v of G⃗. A firefighter starts to protect vertices. At each time interval, the firefighter protects k vertices not yet on fire. Afterward, the fire spreads to all unprotected neighbors that are heads of some arcs starting from the vertices on fire. Let snk(v) denote the maximum number of vertices in G⃗ that the firefighter can save when a fire breaks out at vertex v. The k-surviving rate ρk(G⃗) of G⃗ is defined as ∑v∈V(G⃗)snk(v)/n2.In this paper, we consider the k-surviving rate of digraphs. Main results are as follows: (1) if G⃗ is a k-degenerate digraph, then ρk(G⃗)≥1k+1; (2) if G⃗ is a planar digraph, then ρ2(G⃗)>140; (3) if G⃗ is a planar digraph without 4-cycles, then ρ1(G⃗)>151.