Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth

TL;DR

This paper investigates bounds on the minimum feedback vertex set size in both directed and undirected graphs with bounded degeneracy or treewidth, revealing new tight bounds and constructions for various cases.

Contribution

It establishes new upper and lower bounds for feedback vertex sets in graphs with bounded degeneracy or treewidth, especially highlighting differences between even and odd degeneracy.

Findings

In undirected graphs, bounds are tight for certain degeneracy values.

For even degeneracy, the feedback vertex set size is strictly less than a known upper bound.

In directed graphs, bounds depend on degeneracy and treewidth, with new tight bounds and constructions.

Abstract

We study the minimum size $f$ of a feedback vertex set in directed and undirected $n$-vertex graphs of given degeneracy or treewidth. In the undirected setting the bound $\frac{k-1}{k+1}n$ is known to be tight for graphs with bounded treewidth $k$ or bounded odd degeneracy $k$. We show that neither of the easy upper and lower bounds $\frac{k-1}{k+1}n$ and $\frac{k}{k+2}n$ can be exact for the case of even degeneracy. More precisely, for even degeneracy $k$ we prove that $f < \frac{k}{k+2}n$ and for every $\epsilon>0$, there exists a $k$-degenerate graph for which $f\geq \frac{3k-2}{3k+4}n -\epsilon$. For directed graphs of bounded degeneracy $k$, we prove that $f\leq\frac{k-1}{k+1}n$ and that this inequality is strict when $k$ is odd. For directed graphs of bounded treewidth $k\geq 2$, we show that $f \leq \frac{k}{k+3}n$ and for every $\epsilon>0$, there exists a $k$-degenerate graph for which $f\geq \frac{k-2\lfloor\log_2(k)\rfloor}{k+1}n -\epsilon$. Further, we provide several constructions of low degeneracy or treewidth and large $f$.