Essentially Tight Kernels for (Weakly) Closed Graphs

TL;DR

This paper investigates kernelization for hard graph problems on graphs with triadic closure properties, introducing tight kernel bounds based on closure parameters and extending previous results on degenerate graphs.

Contribution

It provides the first kernelization bounds for several problems based on closure parameters, extending and tightening previous kernelization results for degenerate graphs.

Findings

First kernels of size $k^{\mathcal{O}(\gamma)}$ for Capacitated Vertex Cover

Kernels of size $(\gamma k)^{\mathcal{O}(\gamma)}$ for Connected Vertex Cover

Lower bounds for kernelization of Independent Set on graphs with constant closure number

Abstract

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.