Power Domination and Resolving Power Domination of Fractal Cubic Network

TL;DR

This paper investigates the power domination and resolving power domination parameters of fractal cubic networks, a variant of hypercubes, providing new insights into their monitoring capabilities and structural properties.

Contribution

It introduces the first analysis of power domination and resolving power domination in fractal cubic networks, correcting previous definitions and exploring their unique properties.

Findings

Determined the power dominating set of fractal cubic networks.

Analyzed the resolving power domination set and its challenges.

Compared properties with hypercube networks.

Abstract

In network theory, the domination parameter is vital in investigating several structural features of the networks, including connectedness, their tendency to form clusters, compactness, and symmetry. In this context, various domination parameters have been created using several properties to determine where machines should be placed to ensure that all the places are monitored. To ensure efficient and effective operation, a piece of equipment must monitor their network (power networks) to answer whenever there is a change in the demand and availability conditions. Consequently, phasor measurement units (PMUs) are utilised by numerous electrical companies to monitor their networks perpetually. Overseeing an electrical system which consists of minimum PMUs is the same as the vertex covering the problem of graph theory, in which a subset D of a vertex set V is a power dominating set (PDS) if it monitors generators, cables, and all other components, in the electrical system using a few guidelines. Hypercube is one of the versatile, most popular, adaptable, and convertible interconnection networks. Its appealing qualities led to the development of other hypercube variants. A fractal cubic network is a new variant of the hypercube that can be used as a best substitute in case faults occur in the hypercube, which was wrongly defined in [Eng. Sci. Technol. 18(1) (2015) 32-41]. Arulperumjothi et al. have recently corrected this definition and redefined this variant with the exact definition in [Appl. Math. Comput. 452 (2023) 128037]. This article determines the PDS of the fractal cubic network. Further, we investigate the resolving power dominating set (RPDS), which contrasts starkly with hypercubes, where resolving power domination is inherently challenging.