A Linear-Time Algorithm for Minimum $k$-Hop Dominating Set of a Cactus Graph

TL;DR

This paper introduces a linear-time algorithm for finding minimum $k$-hop dominating sets specifically in cactus graphs, improving efficiency over previous methods and utilizing a reduction to the piercing circular arcs problem.

Contribution

It presents the first linear-time algorithm for minimum $k$-hop dominating sets in cactus graphs, reducing the problem to a circular arcs piercing problem with an optimized solution.

Findings

Linear-time algorithm for cactus graphs.

Reduction to piercing circular arcs problem.

Improved over previous $O(n\log n)$ algorithm.

Abstract

Given a graph $G=(V,E)$ and an integer $k \ge 1$, a $k$-hop dominating set $D$ of $G$ is a subset of $V$, such that, for every vertex $v \in V$, there exists a node $u \in D$ whose hop-distance from $v$ is at most $k$. A $k$-hop dominating set of minimum cardinality is called a minimum $k$-hop dominating set. In this paper, we present linear-time algorithms that find a minimum $k$-hop dominating set in unicyclic and cactus graphs. To achieve this, we show that the $k$-dominating set problem on unicycle graph reduces to the piercing circular arcs problem, and show a linear-time algorithm for piercing sorted circular arcs, which improves the best known $O(n\log n)$-time algorithm.