Kernels for deletion to classes of acyclic digraphs

Given a digraph D and an integer k, Directed Feedback Vertex Set (DFVS) asks whether there exists a set of vertices S of size at most k such that F=D∖S is DAG. Mnich and van Leeuwen [STACS 2016 ] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed light on the kernelization complexity of DFVS, a well-known open problem in Parameterized Complexity. We improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k3) to O(k2) and of Pumpkin Vertex Deletion Set from O(k18) to O(k3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions.