  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.
