  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.
