Comparison of Connected Power Domination Number with Connected Equitable Power Domination Number

Abstract

This paper investigates the strategic placement of monitoring units in networks by exploring the structural relationships between different graph domination parameters. We introduce a comparative analysis between the Connected Power Domination Number (CPDN) and the newly proposed Connected Equitable Power Domination Number (CEPDN) across several distinct graph families, including paths, cycles, stars, wheels, ladders, bipartite graphs, combs, and twig graphs. By establishing exact values for both parameters across these configurations, we evaluate the precise mathematical trade-off between network monitoring efficiency and workload equitability constraints. Our results demonstrate that while equitable constraints match standard propagation efficiency in highly symmetric topologies, they significantly alter monitor requirements in asymmetric frameworks like ladder and twig structures. These findings provide foundational insights for designing balanced, fault-tolerant network monitoring systems.