A cubic vertex-kernel for Trivially Perfect Editing

TL;DR

This paper proves the existence of a cubic vertex-kernel for all variants of the Trivially Perfect Editing problem, improving previous bounds and confirming conjectures about kernel sizes.

Contribution

We establish the first cubic vertex-kernel for Trivially Perfect Editing and its variants, resolving an open problem in parameterized complexity.

Findings

Existence of a cubic vertex-kernel for all variants.

Improved bounds from previous $O(k^7)$ kernels.

Confirmation of conjectured kernel sizes.

Abstract

We consider the Trivially Perfect Editing problem, where one is given an undirected graph $G = (V,E)$ and a parameter $k \in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-Complete and to admit so-called polynomial kernels. More precisely, the existence of an $O(k^3)$ vertex-kernel for Trivially Perfect Completion was announced by Guo (ISAAC 2007) but without a stand-alone proof. More recently, Drange and Pilipczuk (Algorithmica 2018) provided $O(k^7)$ vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem.