A quasi-quadratic vertex Kernel for Cograph edge editing

TL;DR

This paper presents an improved vertex kernel for cograph edge editing with a near quadratic size, utilizing t-modules to generalize previous reduction rules and enhance kernelization efficiency.

Contribution

It introduces t-modules to generalize existing reduction rules, achieving a near quadratic kernel size for cograph edge editing.

Findings

Achieved an $O(k^2 ext{log} k)$ vertex kernel size.

Improved upon previous cubic kernel by Guillemot et al.

Introduced t-modules to facilitate kernel reduction.

Abstract

We provide a $O(k^2 \mathrm{log} k)$ vertex kernel for cograph edge editing. This improves a cubic kernel found by Guillemot, Havet, Paul and Perez [1] which involved four reduction rules. We generalize one of their rules, based on packing of induced paths of length four, by introducing t-modules, which are modules up to t edge modifications. The key fact is that large t-modules cannot be edited more than t times, and this allows to obtain a near quadratic kernel. The extra $\mathrm{log} k$ factor seems tricky to remove as it is necessary in the combinatorial lemma on trees which is central in our proof. Nevertheless, we think that a quadratic bound should be reachable.