Feedback vertex sets of digraphs with bounded maximum degree

TL;DR

The authors address the problem of bounding the size of minimum feedback vertex sets in digraphs with bounded maximum degree, establishing tight upper bounds for Δ≤4 and Δ≤5 for both oriented graphs and general digraphs. Their approach combines structural decompositions, notably a special class H for Δ≤4, and reductions to bipartite subgraphs after forming minimum feedback arc sets for Δ≤5. They prove fvs ≤ 3n/7 for oriented graphs with Δ≤4, fvs ≤ n/2 in several Δ≤4 cases, and fvs ≤ n/2 (or 2n/3) for Δ≤5, with constructions showing tightness. The paper culminates with a conjectured asymptotic behavior f(k) and open questions, guiding future work on fvs bounds in broader Δ regimes.

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

Figures (15)

• Figure 1: The digraphs $H_1$, $H_2$ and $H_3$.
• Figure 2: A digraph in ${\cal H}$ (where the thick arcs are bridges).
• Figure 3: a $2$-regular digraph of order $7$ with feedback vertex set equal to 3
• Figure 4: A subgraph of $D$ in Case E.2.
• Figure 5: The digraph $C_{yx}-z$ in Case E.2.

Theorems & Definitions (12)

• Theorem 2.1
• Corollary 2.2
• Lemma 2.3
• Theorem 2.4
• Claim 1
• Claim 2
• Lemma 3.1
• Claim 3
• Theorem 3.2
• Theorem 3.3