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

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 $R^{\prime }$, would imply that for any $x\in R^{\prime }$ (i.e., fault-detection) $R^{\prime }\setminus \left\{x\right\}$ (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 $di{m}_{m,f}\left(G\right)$ of G. We consider the Cartesian product of $P_2$ and $P_n$ (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 $di{m}_{m,f}$ 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.