Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems
(2005.03986)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 kO(c), that Induced Matching admits a kernel with O(c7*k8) vertices, and that Irredundant Set admits a kernel with O(c5/2*k3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.
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