  Given a graph $G$ and an integer $k$, Max Min FVS asks whether there exists a
minimal set of vertices of size at least $k$ whose deletion destroys all
cycles. We present several results that improve upon the state of the art of
the parameterized complexity of this problem with respect to both structural
and natural parameters.
  Using standard DP techniques, we first present an algorithm of time
$\textrm{tw}^{O(\textrm{tw})}n^{O(1)}$, significantly generalizing a recent
algorithm of Gaikwad et al. of time $\textrm{vc}^{O(\textrm{vc})}n^{O(1)}$,
where $\textrm{tw}, \textrm{vc}$ denote the input graph's treewidth and vertex
cover respectively. Subsequently, we show that both of these algorithms are
essentially optimal, since a $\textrm{vc}^{o(\textrm{vc})}n^{O(1)}$ algorithm
would refute the ETH.
  With respect to the natural parameter $k$, the aforementioned recent work by
Gaikwad et al. claimed an FPT branching algorithm with complexity $10^k
n^{O(1)}$. We point out that this algorithm is incorrect and present a
branching algorithm of complexity $9.34^k n^{O(1)}$.
