# Fault-Tolerant Resolvability and Extremal Structures of Graphs
## 4 Concluding Remarks  
This paper investigates the fault-tolerant metric dimension of graphs. We present certain characterizations of graphs with some extreme values of the fault-tolerant metric dimension. A method is presented to calculate the upper bounds on the fault-tolerant metric dimension of graphs. We study fault-tolerant resolvability in three non-finite families of regular graphs and show that they are the families of graphs with a constant fault-tolerant metric dimension. The following remark shows a comparison between the upper bound produced by our method and the upper bound by [PERSON] et al.  
**Remark 1**.: _Note that the upper bound on the fault-tolerant metric dimension provided by Theorem 4 is always crude. For example, if \(\Gamma\in\{P(n),A(n)\}\) or \(S(n)\), with \(n\equiv 0,2,3(\mod 4)\), then by using \(\beta(\Gamma)=3\) in Theorem 4, we obtain \(\beta^{\prime}(\Gamma)\leq 153\), which is not interesting. In view of this fact, Lemma 3 always gives a much better bound on \(\beta^{\prime}(\Gamma)\)._  
Recently, [PERSON] et al. [36] studied the fault-tolerant metric dimension of hexagonal, honeycomb, and hex-derived networks. See [37] for a study of hexagonal and honeycomb networks. We conclude the paper with some open problems.  
**Problem 1**.: _In view of the characterizations of graphs with fault-tolerant metric dimension \(2\) and \(n-1\), the following open problems are proposed._  
1. _Characterize_ \(n\)_-ordered graphs with fault-tolerant metric dimension_ \(3\)_._
2. _Characterize_ \(n\)_-ordered graphs with fault-tolerant metric dimension_ \(n-2\)_._ _We also propose the following open problems:_  
1. _Study the fault-tolerant metric dimension of other interesting families of the regular graph, such as the prism graphs, and the generalized Petersen graphs_ \(P(n,m)\)_,_ \(m>2\)_._
2. _Investigate the fault-tolerant metric dimension of strongly regular graphs, such as the square grid graphs and the triangular graphs._
3. _In view of [PERSON] et al. [36], study the fault-tolerant resolvability in other direct and multiplex interconnection networks, such as the butterfly and Benes networks._* _Study the applicability of fault-tolerant resolvability in the optimal flow control of multiplex interconnection networks; see, for example,_ _[_38; 39; 40_]_ _for a through study on multiplex networks._  
**Author Contributions:** The idea to study the fault-tolerant resolvability was proposed by [PERSON] After several discussion with [PERSON] and [PERSON], they approved the idea to work on these problems. [PERSON] worked on the problem with assistance from [PERSON]. The Results section was written by [PERSON] and [PERSON]. The Introduction was written by [PERSON], and the Preliminaries section was arranged and written by [PERSON]. The final version was carefully read by [PERSON] and [PERSON]. [PERSON] finalized the write-up of the paper by following suggestions from other authors.  
**Funding:** This research was supported by the Startup Research Grant Program of Higher Education Commission (HEC) Pakistan under Project# 2285 and grant No. 21-2285/SRGP/R&D/HEC/2018 received by [PERSON]. [PERSON] was supported by the Start-up Research Grant 2016 of United Arab Emirates University, Al Ain, United Arab Emirates via Grant No. G00002233 and UPAR Grant of United Arab Emirates University via Grant No. G00002590. APC was covered by [PERSON] who was funded by a Chinese Government Scholarship.  
**Acknowledgments:** The authors are grateful to the anonymous reviewers for a careful reading of this paper and for all their comments, which lead to a number of improvements of the paper.  
**Conflicts of Interest:** The authors declare no conflict of interest.