Scattered packings of cycles
Aistis Atminas, Marcin Kami\'nski, Jean-Florent Raymond

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 and two positive integers and , asks whether contains a collection of cycles that are pairwise at distance at least . This problem generalizes the problem Disjoint Cycles which corresponds to the case . We prove that when parameterized by , , and the maximum degree , the problem Scattered Cycles admits a kernel on vertices. We also provide a -kernel for the case and a -kernel for the case . Our proofs rely on two simple reduction rules and a careful analysis.
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Taxonomy
Topicsgraph theory and CDMA systems · Limits and Structures in Graph Theory · Advanced Graph Theory Research
