Essentially Tight Kernels for (Weakly) Closed Graphs
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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 k^𝒪(γ) and (γ k)^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 k^o(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.
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