  We study reduction rules for Directed Feedback Vertex Set (DFVS) on instances
without long cycles. A DFVS instance without cycles longer than $d$ naturally
corresponds to an instance of $d$-Hitting Set, however, enumerating all cycles
in an $n$-vertex graph and then kernelizing the resulting $d$-Hitting Set
instance can be too costly, as already enumerating all cycles can take time
$\Omega(n^d)$. We show how to compute a kernel with at most $2^dk^d$ vertices
and at most $d^{3d}k^d$ induced cycles of length at most $d$ (which however,
cannot be enumerated efficiently), where $k$ is the size of a minimum directed
feedback vertex set. We then study classes of graphs whose underlying
undirected graphs have bounded expansion or are nowhere dense; these are very
general classes of sparse graphs, containing e.g. classes excluding a minor or
a topological minor. We prove that for such classes without induced cycles of
length greater than $d$ we can compute a kernel with $O_d(k)$ and
$O_{d,\epsilon}(k^{1+\epsilon})$ vertices for any $\epsilon>0$, respectively,
in time $O_d(n^{O(1)})$ and $O_{d,\epsilon}(n^{O(1)})$, respectively. The most
restricted classes we consider are strongly connected planar graphs without any
(induced or non-induced) long cycles. We show that these have bounded treewidth
and hence DFVS on planar graphs without cycles of length greater than $d$ can
be solved in time $2^{O(d)}\cdot n^{O(1)}$. We finally present a new data
reduction rule for general DFVS and prove that the rule together with a few
standard rules subsumes all the rules applied by Bergougnoux et al. to obtain a
polynomial kernel for DFVS[FVS], i.e., DFVS parameterized by the feedback
vertex set number of the underlying (undirected) graph. We conclude by studying
the LP-based approximation of DFVS.
