# Fault-Tolerant Resolvability and Extremal Structures of Graphs  
[PERSON]  
1 School of Mathematical Sciences, Anhui University, Hefei 230601, China; [EMAIL_ADDRESS] (H.R. ); [EMAIL_ADDRESS] (X.-F.P.) [PERSON]  
2 Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Swabi 23460,  
of distance-regular graphs, such as Grassmann graphs [7] and Johnson graphs [8], which have been studied by [PERSON] and others. The metric dimension of Kneser graphs was also studied by [PERSON] at al. [8]. Graphs of group-theoretic interest, such as Cayely digraphs [9] and Cayley graphs generated by certain finite groups [10], have also been studied from the metric dimension viewpoint. The metric dimension and resolving sets of product graphs, such as the Cartesian product of graphs [11] and categorical product of graphs [12], have also been investigated. Certain infinite families generated from wheel graphs have been studied by [PERSON] et al. [13]. The metric dimension of rotationally symmetric convex polytopes (resp. convex polytopes produced by wheel-related graphs) has been studied by [PERSON] et al. [14] (resp. [PERSON] et al. [15]). The question of whether or not the metric dimension is a finite number was answered in [16]. They showed this result by constructing some infinite families of graphs possessing infinite metric dimension. Similar to many other graph-theoretic parameters, the computational complexity of the metric dimension problem was investigated in [17]. Metric dimension has also been generalized and extended by providing more mathematical rigorous general concepts, such as the \(k\)-metric dimension. [PERSON] et al. [18] introduced another concept: fault tolerance in resolvability, which tolerates the removal of any arbitrary vertex and preserves the resolvability status of the underlying set. By considering the vertices in a resolving set as the location for loran/sonar stations, we can say that the location of any such vertex is distinctly measured by its vertex distances from the site of the stations. From this perspective, a fault-tolerant (unique) resolving set is the one which still preserves the property of a resolving set when neglecting any station at a uniquely determined location of a vertex in the resolving set. Consequently, fault-tolerant resolving sets enhance the applicability of classical resolving sets in graphs. In addition, this shows that the fault-tolerant metric dimension possesses applicative superiority over the metric dimension. [PERSON] [19] investigated certain applications by referring to components of a metric basis as sensors. From the fault-tolerant resolvability point of view, if some sensor is lacking in performance and does not convey information efficiently, the system will not have enough information process in order to tackle the thief/intruder/fire, etc. A fault-tolerant resolving set from this perspective deals with this problem by conveying the information efficiently when one of the sensors does not catch the intruder. It turns out that fault tolerance in resolvability has applicative superiority over classical resolvability in graphs. In other words, the fault-tolerant metric dimension has application wherever the metric dimension is applicable. Nevertheless, fault-tolerant resolving sets have not received much attention from researchers. The fault-tolerant metric dimension of certain interesting graphs possessing chemical importance was studied in [20]. Recently, [PERSON] et al. [21; 22] considered certain rotationally symmetric convex polytopes and studied their fault-tolerant metric dimension and binary-locating dominating sets. The reader is referred to [23] for consideration of fault-tolerant resolvability as an optimization problem and its applicative perspective. We also refer the reader to [24; 25; 26; 27; 28] for a study of other interesting graph-theoretic parameters having potential applications in chemistry.