A quasi-quadratic vertex-kernel for Cograph Edge Editing - Crespelle, Christophe; Pellerin, Rémi; Thomassé, Stéphan (Discrete Applied Mathematics, 2024)
We provide a O(k2logk) vertex kernel for cograph edge editing. This improves a cubic kernel found by Guillemot, Havet, Paul and Perez (Guillemot et al., 2010) 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 logk 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.