Analysis of d-Hop Dominating Set Problem for Directed Graph with Indegree Bounded by One

TL;DR

This paper analyzes the computational complexity of the minimum d-hop dominating set problem in directed graphs with in-degree at most one, providing a polynomial-time solution for this specific class of graphs.

Contribution

It introduces a polynomial-time algorithm for solving the minimum d-hop dominating set problem in directed graphs with bounded in-degree, which is generally NP-complete.

Findings

Polynomial-time solution with complexity O(|V_D|^2)

Identifies properties of graphs with in-degree ≤ 1 that enable efficient solving

Provides a practical approach for cluster formation in ad-hoc networks

Abstract

Efficient communication between nodes in ad-hoc networks can be established through repeated cluster formations with designated \textit{cluster-heads}. In this context minimum d-hop dominating set problem was introduced for cluster formation in ad-hoc networks and is proved to be NP-complete. Hence, an exact solution to this problem for certain subclass of graphs (representing an ad-hoc network) can be beneficial. In this short paper we perform computational complexity analysis of minimum d-hop dominating set problem for directed graphs with in-degree bounded by $1$. The optimum solution of the problem can be found polynomially by exploiting certain properties of the graph under consideration. For a digraph $G_{D}=(V_D,E_D)$ an $\mathcal{O}(|V_D|^2)$ solution is provided to the problem.