Feedback vertex sets of digraphs with bounded maximum degree

A digraph D is an oriented graph if D does not have a pair of opposite arcs. The degree of a vertex v of D is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from D makes it acyclic. Let D be a digraph with n vertices and maximum degree $Δ$. We prove the following bounds. If D is an oriented graph, then fvs(D)≤ 3n/7 when Δ≤ 4 and fvs(D)≤ n/2 when Δ≤ 5. If D is a connected digraph, Δ≤ 4 and D is not obtained from an odd undirected cycle by replacing every edge with the pair of opposite arcs with the same endvertices, then fvs(D)≤ n/2. If D is an arbitrary digraph with Δ≤ 5 then fvs(D)≤ 2n/3. Note that all the above bounds are tight.