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 $k$，目标是向 $G$ 中添加或删除最多 $k$ 条边，以获得一个平凡完美图。在近期的工作中，Dumas、Perez 和 Todinca [Algorithmica 2023] 证明了该问题存在一个大小为 $O(k^3)$ 的核。该结果很大程度上依赖于 Drange 和 Pilipczuk [Algorithmica 2018] 所证明的平凡完美模块的大小可以被 $O(k^2)$ 界定的结论。为得到三次顶点核，Dumas、Perez 和 Todinca [Algorithmica 2023] 进一步表明，一种更复杂的结构——所谓的“梳子”——可被缩减至 $O(k^2)$ 个顶点。在本文中，我们证明对于上述两种结构，该界限均可改进至 $O(k)$，从而得到一个大小为 $O(k^2)$ 的核。我们的方法基于一个简单而有力的观察：任何足够大的结构都包含不受影响的顶点，其邻域在大小为 $k$ 的编辑操作后保持不变，这保证了强结构性质。