Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems
Discrete Mathematics
2022-06-22 v2
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.
Cite
@article{arxiv.2005.03986,
title = {Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems},
author = {Tomohiro Koana and Christian Komusiewicz and Frank Sommer},
journal= {arXiv preprint arXiv:2005.03986},
year = {2022}
}