# Feedback Vertex Sets in (Directed) Graphs of Bounded Degeneracy or Treewidth

### 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 $f$.