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 ∆, the problem Scattered Cycles admits a kernel on $24l^2\Delta^l r log(8l^2 ∆^l r)$ vertices. We also provide a (16$l^2 ∆^l$)-kernel for the case r = 2 and a (148∆r log r)-kernel for the case $l$= 1. Our proofs rely on two simple reduction rules and a careful analysis.