  We study kernelization of classic hard graph problems when the input graphs
fulfill triadic closure properties. More precisely, we consider the recently
introduced parameters closure number $c$ and the weak closure number $\gamma$
[Fox et al., SICOMP 2020] in addition to the standard parameter solution size
$k$. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching
we obtain the first kernels of size $k^{\mathcal{O}(\gamma)}$ and $(\gamma
k)^{\mathcal{O}(\gamma)}$, respectively, thus extending previous kernelization
results on degenerate graphs. The kernels are essentially tight, since these
problems are unlikely to admit kernels of size $k^{o(\gamma)}$ by previous
results on their kernelization complexity in degenerate graphs [Cygan et al.,
ACM TALG 2017]. In addition, we provide lower bounds for the kernelization of
Independent Set on graphs with constant closure number~$c$ and kernels for
Dominating Set on weakly closed split graphs and weakly closed bipartite
graphs.
