Kernelization and approximation of distance-$r$ independent sets on nowhere dense graphs

TL;DR

This paper investigates the duality and kernelization complexity of distance-$r$ independent sets and dominating sets in nowhere dense graphs, establishing an almost linear kernel for the independent set problem.

Contribution

It proves the existence of an almost linear kernel for the distance-$r$ independent set problem on nowhere dense graph classes, advancing understanding of their structural and algorithmic properties.

Findings

Established duality between maximum distance-$2r$ independent sets and minimum distance-$r$ dominating sets.

Proved the existence of an almost linear kernel for the distance-$r$ independent set problem.

Analyzed kernelization complexity in nowhere dense graph classes.

Abstract

For a positive integer $r$, a distance-$r$ independent set in an undirected graph $G$ is a set $I\subseteq V(G)$ of vertices pairwise at distance greater than $r$, while a distance-$r$ dominating set is a set $D\subseteq V(G)$ such that every vertex of the graph is within distance at most $r$ from a vertex from $D$. We study the duality between the maximum size of a distance-$2r$ independent set and the minimum size of a distance-$r$ dominating set in nowhere dense graph classes, as well as the kernelization complexity of the distance-$r$ independent set problem on these graph classes. Specifically, we prove that the distance-$r$ independent set problem admits an almost linear kernel on every nowhere dense graph class.