Kernelization of Arc Disjoint Cycle Packing in $$\alpha $$-Bounded Digraphs

In the ARC DISJOINT CYCLE PACKING problem, we are given a simple directed graph (digraph) G, a positive integer k, and the task is to decide whether there exist k arc disjoint cycles. The problem is known to be W[1]-hard on general digraphs parameterized by the standard parameter k. In this paper we show that the problem admits a polynomial kernel on alpha-bounded digraphs. That is, we give a polynomial-time algorithm, that given an instance (D, k) of ARC DISJOINT CYCLE PACKING, outputs an equivalent instance (D', k') of ARC DISJOINT CYCLE PACKING, such that k' <= k and the size of D' is upper-bounded by a polynomial function of k. For any integer alpha >= 1, the class of alpha-bounded digraphs, denoted by D-alpha, contains a digraph D such that the maximum size of an independent set in D is at most alpha. That is, in D, any set of alpha + 1 vertices has an arc with both end-points in the set. For alpha = 1, this corresponds to the well-studied class of tournaments. Our results generalize the recent result by Bessy et al. [MFCS, 2019] about ARC DISJOINT CYCLE PACKING on tournaments.