A Polynomial Kernel for Diamond-Free Editing

TL;DR

This paper presents a polynomial kernel for the diamond-free editing problem, advancing the understanding of kernelization limits for H-free editing problems and providing a simpler kernel for diamond-free edge deletion.

Contribution

It introduces a polynomial kernel for diamond-free editing and a cubic-vertex kernel for diamond-free edge deletion, simplifying previous approaches.

Findings

Polynomial kernel for diamond-free editing problem.

Cubic-vertex kernel for diamond-free edge deletion.

Progress towards a dichotomy in H-free editing kernelization.

Abstract

An $H$-free editing problem asks whether we can edit at most $k$ edges to make a graph contain no induced copy of the fixed graph $H$. We obtain a polynomial kernel for this problem when $H$ is a diamond. The incompressibility dichotomy for $H$ being a 3-connected graph and the classical complexity dichotomy suggest that except for $H$ being a complete/empty graph, $H$-free editing problems admit polynomial kernels only for a few small graphs $H$. Therefore, we believe that our result is an essential step toward a complete dichotomy on the compressibility of $H$-free editing. Additionally, we give a cubic-vertex kernel for the diamond-free edge deletion problem, which is far simpler than the previous kernel of the same size for the problem.