Feedback vertex sets of digraphs with bounded maximum degree

Abstract

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 $\hat{I}”$. We prove the following bounds. If $D$ is an oriented graph, then $fvs(D)\le \frac{3n}{7}$ when $\hat{I}”\le 4$ and $fvs(D)\le \frac{n}{2}$ when $\hat{I}”\le 5$. If $D$ is a connected digraph, $\hat{I}”\le 4$ 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 $fvs(D)\le \frac{n}{2}$. If $D$ is an arbitrary digraph with $\hat{I}”\le 5$ then $fvs(D)\le \frac{2n}{3}.$ Note that all the above bounds are tight.