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

TL;DR

This paper introduces how the concept of c-closure in graphs can be utilized to develop kernelization algorithms for classic graph problems, resulting in size bounds depending on c and the parameter k.

Contribution

It demonstrates the application of c-closure to kernelization, providing new polynomial kernels for Dominating Set, Induced Matching, and Irredundant Set problems.

Findings

Kernel for Dominating Set with size k^O(c)

Kernel for Induced Matching with O(c^7*k^8) vertices

Kernel for Irredundant Set with O(c^(5/2)*k^3) vertices

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.