Scattered packings of cycles

TL;DR

This paper studies the computational problem of finding multiple cycles in a graph that are sufficiently far apart, providing kernelization results that reduce problem size based on parameters like the number of cycles, distance, and maximum degree.

Contribution

The paper introduces new kernelization algorithms for the Scattered Cycles problem, generalizing disjoint cycles, with explicit kernel sizes depending on key parameters.

Findings

Kernel of size 24 ℓ^2 Δ^ℓ r log(8 ℓ^2 Δ^ℓ r) for general parameters

A smaller kernel of size 16 ℓ^2 Δ^ℓ for r=2

A kernel of size 148 Δ r log r for ℓ=1

Abstract

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