On Relaxation of Dominant Sets

TL;DR

This paper introduces a novel polynomial-time algorithm for solving all k-ruling set problems in graphs, extending existing algorithms for minimum dominating sets and maximal independent sets, with applications to proximity-constrained ruling sets.

Contribution

It presents the first known algorithm that efficiently solves all k-ruling set problems alongside minimum dominating set algorithms, including for proximity-constrained cases.

Findings

Algorithm solves all k-ruling set problems with polynomial overhead.

Effective for (,  - 1) ruling sets with proximity constraints.

Works in conjunction with existing dominating set and independent set algorithms.

Abstract

In a graph $G = (V,E)$, a k-ruling set $S$ is one in which all vertices $V$ \ $S$ are at most $k$ distance from $S$. Finding a minimum k-ruling set is intrinsically linked to the minimum dominating set problem and maximal independent set problem, which have been extensively studied in graph theory. This paper presents the first known algorithm for solving all k-ruling set problems in conjunction with known minimum dominating set algorithms at only additional polynomial time cost compared to a minimum dominating set. The algorithm further succeeds for $(\alpha, \alpha - 1)$ ruling sets in which $\alpha > 1$, for which constraints exist on the proximity of vertices v $\in S$. This secondary application instead works in conjunction with maximal independent set algorithms.