We study reduction rules for Directed Feedback Vertex Set (DFVS) on directed graphs 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 \(\Omega (n^d)\) . To the best of our knowledge, the kernelization of DFVS on graphs without long cycles has not been studied in the literature, except for very restricted cases, e.g., for tournaments, in which all induced cycles are of length three. We show that the natural reduction rule to delete all vertices and edges that do not lie on induced cycles cannot be implemented efficiently, that is, it is W[1]-hard (with respect to parameter d) to decide if a vertex or edge lies on an induced cycle of length at most d even on graphs that become acyclic after the deletion of a single vertex or edge. Based on different reduction rules we then 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 every class \(\mathscr {C} \) with bounded expansion there is a function \(f_\mathscr {C} (d)\) such that for graphs \(G\in \mathscr {C} \) without induced cycles of length greater than d we can compute a kernel with \(f_\mathscr {C} (d)\cdot k\) vertices in time \(f_\mathscr {C} (d)\cdot n^{\mathcal {O}(1)}\) . For every nowhere dense class \(\mathscr {C} \) there is a function \(f_\mathscr {C} (d,\varepsilon )\) such that for graphs \(G\in \mathscr {C} \) without induced cycles of length greater than d we can compute a kernel with \(f_\mathscr {C} (d,\varepsilon )\cdot k^{1+\varepsilon }\) vertices for any \(\varepsilon >0\) in time \(f_\mathscr {C} (d,\varepsilon )\cdot n^{\mathcal {O}(1)}\) . The most restricted classes we consider are strongly connected planar graphs without any (induced or non-induced) long cycles. We show that these classes have treewidth \(\mathcal {O}(d)\) and hence DFVS on planar graphs without cycles of length greater than d can be solved in time \(2^{\mathcal {O}(d)}\cdot n^{\mathcal {O}(1)}\) . We finally present a new data reduction rule for general DFVS and prove that the rule together with a few standard rules subsumes all rules applied in the work of 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.
