# The Application of Fault-Tolerant Partition Resolvability in Cycle-Related Graphs
## 1 Introduction and Preliminaries  
Graph theory is one of the significant territories in mathematics whose theoretical ideas have been highly utilized in research areas in computer science, such as data mining, image capturing and networking (see [1]). Utilization of graph theory has proffered significant stimulation in the development of the field, and graph theoretical concepts have led to many interesting real-life problems in the context of the Internet of Things and for routing, fraud detection, customer analysis and scheduling. Graph theory deals with various distance-related parameters, one of which is the metric dimension of graphs. The concept of the MD of a graph was proposed independently by [PERSON] (see [2]) and [PERSON] et al. (see [3]). A generalized version of the MD of a graph as a PD of a graph was proposed by [PERSON] et al. (see [4]). The MD is based on the distances between the vertices, whereas the PD is based on the distances between the vertices and sets containing vertices. Both of these dimensions contribute in different areas, such as network discovery (see [5]), strategies for mastermind games (see [6]), image processing and game theory (see [3,7]). Fault tolerance in resolvability preserves the resolvability status of the system even when one of the machines is lacking in performing its function. All the system developers are focusing on minimizing the probability of failure and maximizing the consistency of the system. Although the electronic systems have been carefully designed, with the passage of time, any of its units can fail to perform. It is compulsory for a fault-tolerant self-stable system that the faulty originally used chain will be replaced by a chain of units havingfault-free units. Thus, a fault-tolerant design enables a system to perform its function even in the occurrence of faults. Consider a connected graph \(\mathrm{Y}\) of the order \(\alpha\) with the vertex set \(V(\mathrm{Y})\) and edge set \(E(\mathrm{Y})\). If two vertices \(\rho,\tau\in V(\mathrm{Y})\), then the distance \(d(\rho,\tau)\) is the length of the shortest path between \(\rho\) and \(\tau\). The distance between a vertex \(\tau\) and \(Q\subseteq V(\mathrm{Y})\) is defined as \(\min\{d(\tau,q)|q\in Q\}\) and is denoted by \(d(\tau,Q)\). For the vertex \(\tau\in V(\mathrm{Y})\), \(N(\tau)\) is the open neighborhood of \(\tau\) in \(\mathrm{Y}\) (i.e., \(N(\tau)=\{y\in V(\mathrm{Y}):y\text{ is adjacent to }\tau\}\), and the closed neighborhood of \(\tau\) is \(N[\tau]=N(\tau)\cup\{\tau\}\)) (see [8]). For an ordered subset \(\Omega=\{\xi_{1},\xi_{2},\ldots,\xi_{\lambda}\}\) of \(V(\mathrm{Y})\), the representation of vertex \(\tau\) with respect to \(\Omega\) is the \(\lambda\) vector \((d(\tau,\xi_{1}),d(\tau,\xi_{2}),\ldots,d(\tau,\xi_{\lambda}))\), denoted by \(r(\tau|\Omega)\). The subset \(\Omega\) is called the resolving set(RS) of \(\mathrm{Y}\) if the representation of \(\tau\) with respect to \(\Omega\) is distinct for all \(\tau\in V(\mathrm{Y})\). The MD of \(\mathrm{Y}\) is defined as \(\min\{|\Omega|:\Omega\text{ is the RS of }\mathrm{Y}\}\) and is denoted by \(\beta(\mathrm{Y})\). [PERSON] et al. studied the MD on sparse graphs and its applications to zero forcing sets (see [9]). [PERSON] et al. studied the MD of the complement of annihilator graphs associated with commutative rings (see [10]). [PERSON] et al. computed the MD of a kayak paddle graph and cycle with a chord graph (see [11]). [PERSON] et al. studied the MD problem of a flower graph (see [12]). The generalized version of the MD as a fault-tolerant metric dimension (FTMD) originated from [PERSON] et al. in 2008 (see [13]). If for every pair of different vertices \(\rho,\tau\in V(\mathrm{Y})\), then there exist at least two vertices \(w_{1},w_{2}\in\Omega\) such that \(d(\rho,w_{l})\
eq d(\tau,w_{l})\) for \(l\in\{1,2\}\). Then, the RS \(\Omega\) of \(V(\mathrm{Y})\) is called fault-tolerant. The FTMD of \(\mathrm{Y}\) is the least number of vertices in an RS \(\Omega\) and is denoted by \(\beta^{\prime}(\mathrm{Y})\).