An improved kernelization algorithm for Trivially Perfect Editing

TL;DR

This paper improves the kernelization algorithm for Trivially Perfect Editing by reducing the kernel size from O(k^3) to O(k^2) vertices, leveraging structural properties of unaffected vertices.

Contribution

The authors present a novel approach that tightens the bounds on key structures, leading to a more efficient kernelization algorithm for the problem.

Findings

Kernel size reduced from O(k^3) to O(k^2) vertices.

Structural properties of unaffected vertices are key to the improvement.

The approach simplifies previous complex reduction techniques.

Abstract

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.