  We study the parameterized and classical complexity of two related problems
on undirected graphs $G=(V,E)$. In Strong Triadic Closure we aim to label the
edges in $E$ as strong and weak such that at most~$k$ edges are weak and $G$
contains no induced $P_3$ with two strong edges. In Cluster Deletion, we aim to
destroy all induced $P_3$s by a minimum number of edge deletions. We first show
that Strong Triadic Closure admits a $4k$-vertex kernel. Then, we study
parameterization by $\ell:=|E|-k$ and show that both problems are
fixed-parameter tractable and unlikely to admit a polynomial kernel with
respect to $\ell$. Finally, we give a dichotomy of the classical complexity of
both problems on $H$-free graphs for all $H$ of order four.
