Scattered packings of cycles

Abstract

We consider the problem Scattered Cycles which, given a graph G and two positive integers r and ℓ, asks whether G contains a collection of r cycles that are pairwise at distance at least ℓ. This problem generalizes the problem Disjoint Cycles which corresponds to the case ℓ=1. We prove that when parameterized by r, ℓ, and the maximum degree Δ, the problem Scattered Cycles admits a kernel on 24ℓ2Δℓrlog⁡(8ℓ2Δℓr) vertices. We also provide a (16ℓ2Δℓ)-kernel for the case r=2 and a (148Δrlog⁡r)-kernel for the case ℓ=1. Our proofs rely on two simple reduction rules and a careful analysis.