  An $H$-free editing problem asks whether we can edit at most $k$ edges to
make a graph contain no induced copy of the fixed graph $H$. We obtain a
polynomial kernel for this problem when $H$ is a diamond. The incompressibility
dichotomy for $H$ being a 3-connected graph and the classical complexity
dichotomy suggest that except for $H$ being a complete/empty graph, $H$-free
editing problems admit polynomial kernels only for a few small graphs $H$.
Therefore, we believe that our result is an essential step toward a complete
dichotomy on the compressibility of $H$-free editing. Additionally, we give a
cubic-vertex kernel for the diamond-free edge deletion problem, which is far
simpler than the previous kernel of the same size for the problem.
