  In an edge modification problem, we are asked to modify at most $k$ edges to
a given graph to make the graph satisfy a certain property. Depending on the
operations allowed, we have the completion problems and the edge deletion
problems. A great amount of efforts have been devoted to understanding the
kernelization complexity of these problems. We revisit several well-studied
edge modification problems, and develop improved kernels for them:
  \begin{itemize}
  \item a $2 k$-vertex kernel for the cluster edge deletion problem,
  \item a $3 k^2$-vertex kernel for the trivially perfect completion problem,
  \item a $5 k^{1.5}$-vertex kernel for the split completion problem and the
split edge deletion problem, and
  \item a $5 k^{1.5}$-vertex kernel for the pseudo-split completion problem and
the pseudo-split edge deletion problem.
  \end{itemize}
  Moreover, our kernels for split completion and pseudo-split completion have
only $O(k^{2.5})$ edges. Our results also include a $2 k$-vertex kernel for the
strong triadic closure problem, which is related to cluster edge deletion.
