In the Trivially Perfect Editing problem one is given an undirected graph $G = (V,E)$ and an integer $k$ and seeks to add or delete at most $k$ edges in $G$ to obtain a trivially perfect graph. In a recent work, Dumas, Perez and Todinca [Algorithmica 2023] proved that this problem admits a kernel with $O(k^3)$ vertices. This result heavily relies on the fact that the size of trivially perfect modules can be bounded by $O(k^2)$ as shown by Drange and Pilipczuk [Algorithmica 2018]. To obtain their cubic vertex-kernel, Dumas, Perez and Todinca [Algorithmica 2023] then showed that a more intricate structure, so-called \emph{comb}, can be reduced to $O(k^2)$ vertices. In this work we show that the bound can be improved to $O(k)$ for both aforementioned structures and thus obtain a kernel with $O(k^2)$ vertices. Our approach relies on the straightforward yet powerful observation that any large enough structure contains unaffected vertices whose neighborhood remains unchanged by an editing of size $k$, implying strong structural properties.

翻译：在平凡完美编辑问题中，给定一个无向图 $G = (V,E)$ 和一个整数 $k$，目标是添加或删除至多 $k$ 条边，使得 $G$ 成为平凡完美图。在近期工作中，Dumas、Perez 和 Todinca [Algorithmica 2023] 证明该问题存在一个包含 $O(k^3)$ 个顶点的核。这一结果严重依赖于 Drange 和 Pilipczuk [Algorithmica 2018] 所证明的平凡完美模块大小可被 $O(k^2)$ 界定的结论。为得到三次顶点核，Dumas、Perez 和 Todinca [Algorithmica 2023] 进一步展示了一种更复杂的结构（称为“梳子”）可被缩减至 $O(k^2)$ 个顶点。在本文中，我们证明上述两种结构的界均可改进至 $O(k)$，从而得到包含 $O(k^2)$ 个顶点的核。我们的方法基于一个简单而有力的观察：任何足够大的结构均包含未受影响的顶点，其邻域在大小为 $k$ 的编辑操作后保持不变，由此推出强结构性性质。