Improved Kernels and Algorithms for Claw and Diamond Free Edge Deletion Based on Refined Observations

TL;DR

This paper introduces a significantly improved kernelization and a faster algorithm for the claw and diamond-free edge deletion problem, enhancing efficiency in graph modification tasks.

Contribution

The paper presents a new kernel of size O(k^3) vertices and O(k^4) edges and an O*(3.792^k) algorithm, advancing previous bounds substantially.

Findings

Kernel size reduced from O(k^{12}) to O(k^3) vertices.

Algorithm runtime improved to O*(3.792^k).

Enhanced efficiency in solving claw and diamond-free edge deletion.

Abstract

In the {claw, diamond}-free edge deletion problem, we are given a graph $G$ and an integer $k>0$, the question is whether there are at most $k$ edges whose deletion results in a graph without claws and diamonds as induced graphs. Based on some refined observations, we propose a kernel of $O(k^3)$ vertices and $O(k^4)$ edges, significantly improving the previous kernel of $O(k^{12})$ vertices and $O(k^{24})$ edges. In addition, we derive an $O^*(3.792^k)$-time algorithm for the {claw, diamond}-free edge deletion problem.