Feedback vertex sets of digraphs with bounded maximum degree

TL;DR

This paper establishes tight bounds on the minimum vertex set removal needed to make digraphs acyclic, based on maximum degree and orientation properties, advancing understanding of feedback vertex sets in directed graphs.

Contribution

It provides new tight bounds for feedback vertex sets in digraphs with bounded maximum degree, including specific results for oriented and connected graphs.

Findings

For oriented graphs with Δ ≤ 4, fvs(D) ≤ 3n/7.

For oriented graphs with Δ ≤ 5, fvs(D) ≤ n/2.

For arbitrary digraphs with Δ ≤ 5, fvs(D) ≤ 2n/3.

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