# Fault-Tolerant Resolvability and Extremal Structures of Graphs
## 2 Preliminaries
### Extended Petersen Graphs
\right.\]  
Again, in this case, \(R\setminus\{z_{2\ell+1}\}\) distinguishes all the vertices in \(\Gamma\) except the following vertices: (i) \(z_{3+i}\) and \(z_{n-i}\) for \(i=0,2,\ldots,2\ell-4\). (ii) \(z_{2\ell}\), \(y_{2\ell+2}\). (iii) \(z_{2\ell+1}\), \(z_{2\ell+5}\), and \(y_{2\ell+3}\). (iv) \(y_{2\ell+4}\) and \(z_{2\ell+6}\). It is easy to see that vertices with same representation in the outer cycle are at different distances from \(z_{2\ell+1}\).\(d(z_{2\ell+1},z_{2\ell})=1\), \(d(z_{2\ell+1},y_{2\ell+2})=2\) and \(d(z_{2\ell+1},z_{2\ell+5})=4\), \(d(z_{2\ell+1},y_{2\ell+3})=2\) and \(d(z_{2\ell+1},y_{2\ell+4})=3\), \(d(z_{2\ell+1},y_{2\ell+6})=5\). The above discussion shows that \(R\) is a resolving set for \(V(\Gamma)\) when \(n\equiv 3(\mod 4)\) and \(n\geq 11\). Hence, \(\beta(\Gamma)\leq 3\) for \(n\equiv 3(\mod 4)\). **Claim 2:**\(\Gamma\) has a fault-tolerant resolving set of cardinality \(12\) with \(n\geq 11\). We can write \(n=4\ell+3\), \(\ell\geq 2\), \(\ell\in\mathbb{Z}^{+}\). Note that, in this case, \(R=\{y_{1},y_{2},z_{2\ell+1}\}\) is a resolving set of \(\Gamma\). We show that \(\Gamma\) has a fault-tolerant resolving set of cardinality \(12\). From Figure 3, it can be observed that \(N[y_{1}]=\{y_{1},y_{3},y_{n-1},z_{1}\}\), \(N[y_{2}]=\{y_{2},y_{4},y_{n},z_{2}\}\), and \(N[z_{2\ell+1}]=\{z_{2\ell},z_{2\ell+1},z_{2\ell+2},y_{2\ell+1}\}\). Moreover, we find that \(\lambda(N_{\Gamma}(y_{1}))=\lambda(N_{\Gamma}(y_{2}))=\lambda(N_{\Gamma}(z_{2 \ell+1}))=\emptyset\). Thus, by using Lemma 3, we find that \(R^{\prime}=\{y_{1},y_{2},y_{3},y_{4},y_{n-1},y_{n},y_{2\ell+1},z_{1},z_{2},z_{ 2\ell},z_{2\ell+1},z_{2\ell+2}\}\) is a fault-tolerant resolving set of \(P_{n,2}\). Thus, a fault-tolerant resolving set of \(P_{n,2}\) with cardinality \(12\) exists. By using Proposition 1, the above discussion, and Inequality (1), we find that \(\beta^{\prime}(\Gamma)\geq 4\). As a consequence of Theorem 9, we have the following corollary. It provides a fault-tolerant metric dimension analog of Proposition 2. **Corollary 1**. : _The extended Petersen graph \(P(n)\) is a family of graphs with a constant fault-tolerant metric dimension._Proof. : By Theorem 9, we have  
\[4\leq\beta^{\prime}\big{(}P(n)\big{)}\leq\left\{ \begin{array}{ll}10,&\text{if }n \equiv 0(\mod 2)\text{ with }n\geq 8;\\ 12,&\text{if }n\equiv 1(\mod 2)\text{ with }n\geq 11.\end{array} \right.\]  
This implies that the fault-tolerant metric dimension of \(P(n)\) does not depend on the defining parameter \(n\). Thus, by definition, \(P(n)\) is a family of graphs with a constant fault-tolerant metric dimension. In view of Lemma 3 and Proposition 1, we find enough reasoning to propose the following conjecture on the greatest lower bound of the fault-tolerant metric dimension for the extended Petersen graph \(P(n)\). **Conjecture 1**. : _Let \(\Gamma\) be the extended Petersen graph \(P(n)\); then,_  
\[\beta^{\prime}(\Gamma)\geq\left\{ \begin{array}{ll}10,&\text{if }n\equiv 0(\mod 2) \text{ with }n\geq 8;\\ 12,&\text{if }n\equiv 1(\mod 2)\text{ with }n\geq 11,\end{array} \right.\]  
_and thus, we have_  
\[\beta^{\prime}(\Gamma)=\left\{ \begin{array}{ll}10,&\text{if }n\equiv 0(\mod 2) \text{ with }n\geq 8;\\ 12,&\text{if }n\equiv 1(\mod 2)\text{ with }n\geq 11.\end{array} \right.\]