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 γ
[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 kO(γ) and (γk)O(γ), respectively, thus extending previous kernelization
results on degenerate graphs. The kernels are essentially tight, since these
problems are unlikely to admit kernels of size ko(γ) 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