Polynomial Kernels for Paw-free Edge Modification Problems

TL;DR

This paper proves that certain graph modification problems to eliminate paw subgraphs admit polynomial kernels, providing efficient preprocessing algorithms with size bounds proportional to the parameter k.

Contribution

It establishes polynomial kernelizations for paw-free edge modification problems, specifically for completion and deletion, advancing the understanding of problem compressibility.

Findings

Polynomial kernels of O(k) vertices for paw-free completion.

Polynomial kernels of O(k^4) vertices for paw-free edge deletion.

Progress in understanding the compressibility of H-free edge modification problems.

Abstract

Let $H$ be a fixed graph. Given a graph $G$ and an integer $k$, the $H$-free edge modification problem asks whether it is possible to modify at most $k$ edges in $G$ to make it $H$-free. Sandeep and Sivadasan (IPEC 2015) asks whether the paw-free completion problem and the paw-free edge deletion problem admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, $O(k)$-vertex and $O(k^4)$-vertex kernels for them. This is part of an ongoing program that aims at understanding compressibility of $H$-free edge modification problems.