Edge-Version of Fault-Tolerant Resolvability in Networks

Fault tolerance refers to a system’s capacity to continue functioning as intended, even when one of its components fails. Such a system is known as a fault-tolerant, self-stable system. The idea of fault-tolerant resolving sets (FTRS) arises from the concept that removing any vertex from a resolving set (RS) still results in another RS, hence designated as an FTRS. The minimum size of this set is called the fault-tolerant metric dimension (FTMD). This paper extends the concept to edges by introducing the edge version of the fault-tolerant resolving set (EVFTRS) and its corresponding edge version of the fault-tolerant metric dimension (EVFTMD), which is based on edge distances in the network analysis. We calculate the EVFTMD values for n-sunlet and cycle with chord networks, demonstrating that these values remain constant. These findings illustrate the reliability of these network topologies in environments prone to edge failures, offering valuable insights for designing resilient communication systems such as optical networks and smart grids. By adopting an edge-based perspective, this study advances fault tolerance analysis in graph theory and practical network design.