Data Reduction for Directed Feedback Vertex Set on Graphs Without Long Induced Cycles

Abstract

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 $\varOmega (n^d)$. 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 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). 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 $\mathcal {O}_d(k)$ and $\mathcal {O}_{d,\varepsilon }(k^{1+\varepsilon })$ vertices for any $\varepsilon >0$, respectively, in time $\mathcal {O}_d(n^{\mathcal {O}(1)})$ and $\mathcal {O}_{d,\varepsilon }(n^{\mathcal {O}(1)})$, respectively, where k is the size of a minimum directed feedback vertex set. The most restricted classes we consider are planar graphs without any (induced or non-induced) long cycles. We show that strongly connected planar graphs without long cycles have bounded treewidth and hence DFVS on such graphs 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 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.