# Fault-Tolerant Resolvability and Extremal Structures of Graphs
## 2 Preliminaries
### Squared Cycle Graphs
\begin{array}{ll}(k,k-1,k-1),&2\leq k\leq\ell+1;\\ (2\ell-k+2,2\ell-k+3,2\ell-k+3),&\ell+2\leq k\leq 2\ell+1.\end{array} \right.\]  
and  
\[C_{R}(y_{2k+1})=\left\{ \begin{array}{ll}(k,k,k-1),&2\leq k\leq\ell+1;\\ (2\ell-k+2,2\ell-k+2,2\ell-k+3),&\ell+2\leq k\leq 2\ell+1.\end{array} \right.\]Once again, we can see that all vertices in \(\Gamma\) have a distinct representation, which shows that \(\beta(\Gamma)\leq 3\) when \(n\equiv 3(\mod 4)\). **Claim 2:**\(\Gamma\) has a fault-tolerant resolving set of cardinality \(7\) when \(n\geq 7\). We can write \(n=4\ell\), \(\ell\geq 2\), \(\ell\in\mathbb{Z}^{+}\). Note that, in this case, \(R=\{y_{1},y_{2},y_{3}\}\) is a resolving set of \(\Gamma\). We show that \(\Gamma\) is the graph in which there exists a fault-tolerant resolving set of cardinality \(7\). From Figure 5, it can be observed that \(N[y_{1}]=\{y_{1},y_{2},y_{3},y_{n-1},y_{n}\}\), \(N[y_{2}]=\{y_{1},y_{2},y_{3},y_{4},y_{n}\}\), and \(N[y_{3}]=\{y_{1},y_{2},y_{3},y_{4},y_{5}\}\). Moreover, we find that \(\lambda(N_{\Gamma}(y_{1}))=\lambda(N_{\Gamma}(y_{2}))=\lambda(N_{\Gamma}(y_{3 }))=\emptyset\). Thus, by using Lemma 3, we find that \(R^{\prime}=\{y_{1},y_{2},y_{3},y_{4},y_{5},y_{n-1},y_{n}\}\) is a fault-tolerant resolving set of \(\Gamma\). Thus, it is shown that a fault-tolerant resolving set of \(\Gamma\) with cardinality \(7\) exists when \(n\geq 7\). **Claim 1:** There exists a resolving set \(R\) of order \(4\) in \(\Gamma\). **Case 1:** When \(n\equiv 1(\mod 4)\). Now, we can write \(n=4\ell+1\), \(\ell\in\mathbb{Z}^{+}\). \[C_{R}(y_{2k})=\left\{ \begin{array}{ll}(k,k-1,k-1),&2\leq k\leq\ell;\\ (\ell,\ell,\ell),&k=\ell+1;\\ (2\ell-k+1,2\ell-k+2,2\ell-k+2),&\ell+2\leq k\leq 2\ell.\end{array} \right.\]  
and  
\[C_{R}(y_{2k+1})=\left\{ \begin{array}{ll}(k,k,k-1),&2\leq k\leq\ell;\\ (\ell,\ell,\ell),&k=\ell+1;\\ (2\ell-k+1,2\ell-k+1,2\ell-k+2),&\ell+2\leq k\leq 2\ell.\end{array} \right.\]  
For \(n\equiv 1(\mod 4)\), the vertices \(y_{2\ell+2}\) and \(y_{2\ell+3}\) have the same representation. In order to have distinct representations, we add \(y_{2\ell+2}\) to the resolving set \(R\). Now, \(R^{\prime}=\{y_{1},y_{2},y_{3},y_{2\ell+2}\}\) resolves \(V(\Gamma)\). So, it is shown that \(\beta(\Gamma)\leq 4\) for \(n\equiv 1(\mod 4)\). **Claim 2:** When \(n\geq 13\), \(\Gamma\) has a fault-tolerant resolving set of cardinality \(12\). It can be written \(n=4\ell+1\), \(\ell\geq 3\), \(\ell\in\mathbb{Z}^{+}\). Now, for this, \(R=\{y_{1},y_{2},y_{3},y_{2\ell+2}\}\) is a resolving set of \(\Gamma\). We show that \(\Gamma\) has a fault-tolerant resolving set of cardinality \(12\). From Figure 5, it can be observed that \(N[y_{1}]=\{y_{1},y_{2},y_{3},y_{n-1},y_{n}\}\), \(N[y_{2}]=\{y_{1},y_{2},y_{3},y_{4},y_{n}\}\), \(N[y_{3}]=\{y_{1},y_{2},y_{3},y_{4},y_{5}\}\), and \(N[y_{2\ell+2}]=\{y_{2\ell},y_{2\ell+1},y_{2\ell+2},y_{2\ell+3},y_{2\ell+4}\}\). Moreover, we find that \(\lambda(N_{\Gamma}(y_{1}))=\lambda(N_{\Gamma}(y_{2}))=\lambda(N_{\Gamma}(y_{3 }))=\lambda(N_{\Gamma}(y_{2\ell+2}))=\emptyset\). Thus, by using Lemma 3, we find that \(R^{\prime}=\{y_{1},y_{2},y_{3},y_{4},y_{5},y_{n-1},y_{n},y_{2\ell},y_{2\ell+1},y_{2\ell+2},y_{2\ell+3},y_{2\ell+4}\}\) is a fault-tolerant resolving set of \(\Gamma\). Thus, \(\Gamma\) is the graph in which there exists a fault-tolerant resolving set of cardinality \(12\) when \(n\geq 13\). In view of Lemma 3 and Proposition 1, we find enough reasoning to propose the following conjecture on the lower bound of the fault-tolerant metric dimension for the squared cycle graph \(S(n)\). From the above discussion, Inequality, and Proposition 6, we find that  
\[\beta^{\prime}(\Gamma)\geq\left\{ \begin{array}{ll}4,&if\ n\equiv 0,2,3(\mod 4) \ with\ n\geq 5;\\ \delta,&if\ n\equiv 1(\mod 4)\ with\ n\geq 5.\end{array} \right.\]  
where  
\[\delta=\left\{ \begin{array}{ll}4,&if\ \beta(\Gamma)>2;\\ 5,&if\ \beta(\Gamma)>3.\end{array} \right.\]  
Because of Theorem 11, the following corollary is presented. It provides a fault-tolerant metric dimension analogous to Proposition 7.