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

Abstract

In the Arc Disjoint Cycle Packing problem, we are given a 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. 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 \(k'\le k\) and the size of \(D'\) is upper bounded by a polynomial function of k. For any integer \(\alpha \ge 1\), the class of \(\alpha \)-bounded digraphs, denoted by \(\mathcal{D}_\alpha \), 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 generalizes the recent result by Bessy et al. [MFCS 2019] about Arc Disjoint Cycle Packing on tournaments.