A Polynomial Kernel for Trivially Perfect Editing

TL;DR

This paper presents a polynomial kernel of size O(k^7) for Trivially Perfect Editing, resolving an open question and extending kernelization results to related problems, while also establishing complexity bounds.

Contribution

It provides the first polynomial kernel for Trivially Perfect Editing and related problems, and proves a lower bound on its computational complexity under ETH.

Findings

Kernel with O(k^7) vertices for Trivially Perfect Editing.

Polynomial kernels for Trivially Perfect Completion and Deletion.

No subexponential time algorithm for Trivially Perfect Editing unless ETH fails.

Abstract

We give a kernel with $O(k^7)$ vertices for Trivially Perfect Editing, the problem of adding or removing at most $k$ edges in order to make a given graph trivially perfect. This answers in affirmative an open question posed by Nastos and Gao, and by Liu et al. Our general technique implies also the existence of kernels of the same size for the related problems Trivially Perfect Completion and Trivially Perfect Deletion. Whereas for the former an $O(k^3)$ kernel was given by Guo, for the latter no polynomial kernel was known. We complement our study of Trivially Perfect Editing by proving that, contrary to Trivially Perfect Completion, it cannot be solved in time $2^{o(k)} \cdot n^{O(1)}$ unless the Exponential Time Hypothesis fails. In this manner we complete the picture of the parameterized and kernelization complexity of the classic edge modification problems for the class of trivially perfect graphs.