Exploiting $c$-Closure in Kernelization Algorithms for Graph Problems

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 $c$ such that $G$ is $c$-closed. Fox et al. [SIAM J. Comput., 49 (2020), pp. 448--464] 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 $k^{\mathcal{O}(c)}$, that Induced Matching admits a kernel with $\mathcal{O}(c^7 k^{8})$ vertices, and that Irredundant Set admits a kernel with $\mathcal{O}(c^{5/2} k^3)$ vertices. As we show, our kernelizations exploit the fact that $c$-closed graphs have polynomially bounded Ramsey numbers.