A Cubic Vertex-Kernel for Trivially Perfect Editing

We consider the Trivially Perfect Editing problem, where one is given an undirected graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V,E)$$\end{document} and a parameter k∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \in {\mathbb {N}}$$\end{document} 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 (Burzyn et al., in Discret Appl Math 154(13):1824–1844, 2006; Nastos and Gao, in Soc Netw 35(3):439–450, 2013) and to admit so-called polynomial kernels (Drange and Pilipczuk, in Algorithmica 80(12):3481–3524, 2018; Guo, in: Tokuyama, (ed) Algorithms and Computation, 18th International Symposium, ISAAC. Lecture Notes in Computer Science, Springer, Sendai, 2007. https://doi.org/10.1007/978-3-540-77120-3_79; Bathie et al., in Algorithmica 1–27, 2022). More precisely, Drange and Pilipczuk (Algorithmica 80(12):3481–3524, 2018) provided O(k7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(k^7)$$\end{document} 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. Notice that a quadratic vertex-kernel was recently obtained for Trivially Perfect Completion by Bathie et al. (Algorithmica 1–27, 2022).