Kernelization of Arc Disjoint Cycle Packing in α -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 α -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 α ≥ 1, the class of α -bounded digraphs, denoted by 𝒟_α , contains a digraph D such that the maximum size of an independent set in D is at most α . That is, in D , any set of α + 1 vertices has an arc with both end-points in the set. For α = 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.