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

Abstract: In the {claw, diamond}-free edge deletion problem, we are given a graph $G$ and an integer $k&gt;0$, 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 $O(k<sup>3)$ vertices and $O(k<sup>4)$ edges, significantly improving the previous kernel of $O(k<sup>{12})$ vertices and $O(k<sup>{24})$ edges. In addition, we derive an $O<sup>*(3.792<sup>k)$-time algorithm for the {claw, diamond}-free edge deletion problem.