Fault tolerance in para-line network topologies: theory and applications in smart systems.

This study explores the fault-tolerant metric dimension (FTMD) of the para-line network, which is derived from the n-sunlet network, a significant category of networks created by subdividing and transforming cycle-based networks to simulate greater structural complexity. FTMD ensures robust vertex identification even when some nodes fail, a characteristic essential for fault-resilient systems like communication and sensor networks. By rigorously analyzing the structural attributes of these para-line networks, we demonstrate that the FTMD consistently behaves in a specific manner: it is equal to 3 when [Formula: see text] or 6, and 4 otherwise. Our approach includes constructive distance-vector analysis and combinatorial proofs to ensure the smallest possible size of the fault-tolerant resolving set (FTRS). These findings contribute both theoretically and practically, offering new perspectives on the development of resilient network topologies for practical applications such as IoT infrastructure, smart cities, and fault-tolerant distributed systems.