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arXiv · 2005.03986

Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems

Tomohiro Koana·Christian Komusiewicz·Frank Sommer
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Abstract

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.

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Tomohiro Koana, Christian Komusiewicz, Frank Sommer. 2020-05-08. Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems. https://arxiv.org/abs/2005.03986

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