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.

中文翻译：

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三次顶点内核，可轻松进行完美编辑

我们考虑平凡完美的编辑问题，在其中给出一个无向图$ G =（V，E）$和一个参数$ k \ in \ mathbb {N} $，并试图编辑（添加或删除）$ k从$ G $的$边获得平凡的理想图形。通过分别仅允许边添加或边删除来获得相关的平凡完美完成和平凡完美删除问题。平凡的图既是弦图又是音图，并且具有与树深度宽度参数和社交网络分析有关的应用。已知该问题的所有变体都是NP完全的，并且可以接受所谓的多项式核。更准确地说，郭（ISAAC 2007）宣布存在一个用于平凡完美完成的$ O（k ^ 3）$顶点内核，但没有独立的证明。最近，Drange和Pilipczuk（Algorithmica 2018）为这些问题提供了$ O（k ^ 7）$个顶点内核，并开放了立方顶点内核的存在。在这项工作中，对于该问题的所有三个变体，我们都对该问题做出肯定的回答。