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

Efficient communication between nodes in ad-hoc networks can be established through repeated cluster formations with designated $\mathit{\text{cluster-heads}}$. 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 $\mathcal{O}(\mathrm{\mid }{V}_D{\mathrm{\mid }}^2)$ solution is provided to the problem.