Scattered packings of cycles

We consider the problem Scattered Cycles which, given a graph G and two positive integers r and `, asks whether G contains a collection of r cycles that are pairwise at distance at least `. This problem generalizes the problem Disjoint Cycles which corresponds to the case ` = 1. We prove that when parameterized by r, `, and the maximum degree ∆, the problem Scattered Cycles admits a kernel on 24`∆r log(8`∆r) vertices. We also provide a (16`∆)-kernel for the case r = 2 and a (148∆r log r)-kernel for the case ` = 1. Our proofs rely on two simple reduction rules and a careful analysis.