  The Directed Feedback Vertex Set (DFVS) problem takes as input a directed
graph~$G$ and seeks a smallest vertex set~$S$ that hits all cycles in $G$. This
is one of Karp's 21 $\mathsf{NP}$-complete problems. Resolving the
parameterized complexity status of DFVS was a long-standing open problem until
Chen et al. [STOC 2008, J. ACM 2008] showed its fixed-parameter tractability
via a $4^kk! n^{\mathcal{O}(1)}$-time algorithm, where $k = |S|$.
  Here we show fixed-parameter tractability of two generalizations of DFVS:
  - Find a smallest vertex set $S$ such that every strong component of $G - S$
has size at most~$s$: we give an algorithm solving this problem in time
$4^k(ks+k+s)!\cdot n^{\mathcal{O}(1)}$. This generalizes an algorithm by Xiao
[JCSS 2017] for the undirected version of the problem.
  - Find a smallest vertex set $S$ such that every non-trivial strong component
of $G - S$ is 1-out-regular: we give an algorithm solving this problem in time
$2^{\mathcal{O}(k^3)}\cdot n^{\mathcal{O}(1)}$.
  We also solve the corresponding arc versions of these problems by
fixed-parameter algorithms.
