Advanced Mathematical Models & Applications

This paper investigates the certified domination number γ_cer (G), a robust graph parameter, for various graph structures. We determine γ_cer (G) for fundamental graphs like paths (P_n) and cycles (C_n). We mainly contribute by rigorously deriving the certified domination number for various distance graphs and shadow distance graphs, which are created from traditional graph families like pathways and cycles. We investigate these specialized graph classes and find that the behavior and complexity of certified domination are affected by modest structural alterations. Specifically, our findings elucidate the complex interactions among vertex connectivity, distance limitations, and domination certification, unveiling previously unexplored boundaries and characterizations. When dependability and verifiability are crucial in real-world situations, these insights are extremely important. Our findings have ramifications for various application sectors. Certified dominance in robust sensor network architecture guarantees coverage even when there are errors or hostile circumstances. It ensures that monitoring configurations for verifiable surveillance systems can be independently verified. Resilient topologies that are impervious to intrusion or compromise can be built upon the certified dominance structures that underlie secure communication protocols. Our approach establishes a foundational framework for further investigation of domination-based metrics in dynamic and limited situations by linking theoretical graph characteristics with applied system needs.