The surviving rate of digraphs

Let G ⃗ be a connected digraph 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 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 sn k ( v ) 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 ρ k ( G ⃗ ) of G ⃗ is defined as ∑ v ∈ V ( G ⃗ ) sn k ( v ) / n 2 . s paper, we consider the k -surviving rate of digraphs. Main results are as follows: (1) if G ⃗ is a k -degenerate digraph, then ρ k ( G ⃗ ) ≥ 1 k + 1 ; (2) if G ⃗ is a planar digraph, then ρ 2 ( G ⃗ ) > 1 40 ; (3) if G ⃗ is a planar digraph without 4-cycles, then ρ 1 ( G ⃗ ) > 1 51 .