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

Abstract

Given a graph G = (V, E) and an integer k ≥ 1, a k-hop dominating set D of G is a subset of V , such that, for every vertex v ∈ V , there exists a node u ∈ 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.