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arXiv · 2012.05869

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

A. Karim Abu-Affash·Paz Carmi·Adi Krasin
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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.

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A. Karim Abu-Affash, Paz Carmi, Adi Krasin. 2020-12-10. A Linear-Time Algorithm for Minimum $k$-Hop Dominating Set of a Cactus Graph. https://arxiv.org/abs/2012.05869

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