  We give a kernel with $O(k^7)$ vertices for Trivially Perfect Editing, the
problem of adding or removing at most $k$ edges in order to make a given graph
trivially perfect. This answers in affirmative an open question posed by Nastos
and Gao, and by Liu et al. Our general technique implies also the existence of
kernels of the same size for the related problems Trivially Perfect Completion
and Trivially Perfect Deletion. Whereas for the former an $O(k^3)$ kernel was
given by Guo, for the latter no polynomial kernel was known.
  We complement our study of Trivially Perfect Editing by proving that,
contrary to Trivially Perfect Completion, it cannot be solved in time $2^{o(k)}
\cdot n^{O(1)}$ unless the Exponential Time Hypothesis fails. In this manner we
complete the picture of the parameterized and kernelization complexity of the
classic edge modification problems for the class of trivially perfect graphs.
