A linear-time algorithm for minimum k-hop dominating set of a cactus graph

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 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 a linear-time algorithm that finds a minimum k-hop dominating set in cactus graphs, which improves the O(n³)-time algorithm of Borradaile and Le (2017). To achieve this, we show that the k-hop dominating set problem for unicyclic graphs reduces to the piercing circular arcs problem, and show a linear-time algorithm for piercing sorted circular arcs, which improves the best known O(nlogn)-time algorithm.