Fault-tolerance and unique identification of vertices and edges in a graph: The fault-tolerant mixed metric dimension

• Published in: Journal of parallel and distributed computing Vol. 197; p. 105024
• Main Authors: Khan, Asad, Ali, Sikander, Hayat, Sakander, Azeem, Muhammad, Zhong, Yubin, Zahid, Manzoor Ahmad, Alenazi, Mohammed J.F.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2025
• Subjects: Cartesian product; Fault-tolerance; Ladder network; Resolving set; Vertex-edge locating set; Resolving set; Ladder network; 05C12; Vertex-edge locating set; 05C85; 05C90; Fault-tolerance; Cartesian product
• ISSN: 0743-7315
• DOI: 10.1016/j.jpdc.2024.105024

The practical and theoretical significance of graph-theoretic resolvability/locating parameters make them important tools, particularly in the context of network analysis. Their significance is seen in diverse scientific fields and various applications including network security, facility location, efficient routing, social network analysis, and the optimization of site selection. In order to enhance the practical applicability of vertex-edge resolvability in graphs, this paper introduces fault-tolerance in it and studies the minimality of this vertex-edge fault-tolerant resolving sets in graphs. Let R be a set that serves as both a locating and an edge-locating (i.e., mixed locating set) in graph G, implying that it uniquely identifies both vertices and edges in G. Introduction of fault-tolerance in R, say R′, would imply that for any x∈R′ (i.e., fault-detection) R′∖{x} (i.e., fault-tolerance) retains its status of a fault-tolerant mixed locating set. The smallest cardinality of a fault-tolerant mixed locating set is named as the fault-tolerant mixed metric dimension dimm,f(G) of G. We consider the Cartesian product of P2 and Pn (n-dimensional path graph) which is also called the ladder network and deliver its applications in electrical, electronics, and wireless communication areas. We compute the exact value of dimm,f for the ladder network and deliver its potential applications. The exchange property corresponding to the fault-tolerant mixed metric dimension for the ladder networks is also investigated. •The notion of the fault-tolerant mixed metric dimension of networks is studied.•The fault-tolerant mixed metric dimension enhances applicability of the traditional mixed metric dimension.•Applications to ladder networks are delivered to justify their consideration.•The exact value of the fault-tolerant mixed metric dimension for the ladder networks is computed to be 4.•Exchange property regarding fault-tolerant mixed resolvability in ladder networks is satisfied.