  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 $f$.
