# Fault-Tolerant Resolvability and Extremal Structures of Graphs
## 2 Preliminaries
### Squared Cycle Graphs  
[PERSON] et al. [32] proved that the fault-tolerant metric dimension of cycle graphs is 3. Lemma 4: _Let \(\Gamma\) be the cycle graph \(C_{n}\), where \(n\geq 3\). Then, \(\beta^{\prime}(\Gamma)=3\)._  
In the subsequent section, we study fault-tolerant resolvability of squared cycle graphs, which are somewhat of an extension of cycle graphs. The squared cycle graph \(S(n)\) is a 4-regular graph of order \(n\), with \(V(S(n))=\{y_{1},y_{2},\ldots,y_{n}\}\). For each \(k\)\((1\leq k\leq n)\), we join \(y_{k}\) to \(y_{k+1},y_{k+2}\) and to \(y_{k-1},y_{k-2}\). If we cyclically arrange the vertices \(y_{1},y_{2},\ldots,y_{n}\), then each vertex \(y_{k}\) is adjacent to the 2 vertices that immediately follow \(y_{k}\) and 2 vertices that immediately precede \(y_{k}\). Thus, \(S(n)\) is a _four_-regular graph. In Figure 5, we depict the squared cycle graph \(S(n)\) for \(n=8\) and \(n=9\). Note that the squared cycle graph is a special case of the Harary graph \(H(m,n)\), with \(m=4\). [PERSON] et al. [34] showed the following result. Proposition 6: [34] _For \(n\equiv 0,2,3(\mod 4)\), let \(\Gamma\) be the squared cycle graph \(S(n)\) with \(n\geq 5\). Then, \(\beta(\Gamma)=3\)._  
They also showed the following:  
Proposition 7: [34] _For a positive integer \(n\), the squared cycle graph \(S(n)\) is a family of graphs with a constant metric dimension._  
The following is the main result of this section. Note that Claim 1 in the following result was essentially shown in Proposition 6. Theorem 11: _Let \(\Gamma\) be the squared cycle graph \(S(n)\); then,_  
\[\beta^{\prime}(\Gamma)\leq\left\{ \begin{array}{ll}7,&if\ n\equiv 0,2,3(\mod 4) \ with\ n\geq 7;\\ 12,&if\ n\equiv 1(\mod 4)\ with\ n\geq 13,\end{array} \right.\]  
Figure 5: (**a**) The squared cycle graph \(S(8)\), (**b**) the squared cycle graph \(S(9)\). _and_  
\[\beta^{\prime}(\Gamma)\geq\left\{ \begin{array}{ll}4,&if\ n\equiv 0,2,3(\mod 4)\ with\ n\geq 5; \\ \delta,&if\ n\equiv 1(\mod 4)\ with\ n\geq 5.\end{array} \right.\]  
_where_  
\[\delta=\left\{ \begin{array}{ll}4,&if\ \beta(\Gamma)>2;\\ 5,&if\ \beta(\Gamma)>3.\end{array} \right.\]  
Proof. : Let \(\Gamma\) be the squared cycle graph \(S(n)\). We show the following claims to complete the proof. **Claim 1:** There exists a resolving set \(R\) of order \(3\) in \(\Gamma\). **Case 1:** When \(n\equiv 0,2,3(\mod 4)\) with \(n\geq 7\). Based on the location of basis elements in \(\Gamma\), we further divide this case into three subcases. **Subcase 1.1:** When \(n\equiv 0(\mod 4)\). We can write \(n=4\ell\), \(\ell\in\mathbb{Z}^{+}\). We prove that \(R=\{y_{1},y_{2},y_{3}\}\) resolves \(V(\Gamma)\). In order to show that \(R\) resolves vertices of \(V(\Gamma)\), the representation of the vertices of \(V(\Gamma)\) with respect to the resolving set is given. \[C_{R}(y_{2k})=\left\{ \begin{array}{ll}(k,k-1,k-1),&2\leq k\leq\ell;\\ (\ell,\ell,\ell),&k=\ell+1;\\ (2\ell-k+1,2\ell-k+1,2\ell-k+2),&\ell+2\leq k\leq 2\ell.\end{array} \right.\]  
and  
\[C_{R}(y_{2k+1})=\left\{ \begin{array}{ll}(k,k,k-1),&2\leq k\leq\ell;\\ (2\ell-k,2\ell-k+1,2\ell-k+1),&\ell+1\leq k\leq 2\ell-1.\end{array} \right.\]  
From the above discussion, it is shown that all vertices have a distinct representation for \(n\equiv 0(\mod 4)\), so \(\beta(G)\leq 3\). **Subcase 1.2:** When \(n\equiv 2(\mod 4)\). It can be written as \(n=4\ell+2\), \(\ell\in\mathbb{Z}^{+}\). \[C_{R}(y_{2k})=\left\{ \begin{array}{ll}(k,k-1,k-1),&2\leq k\leq\ell+1;\\ (2\ell-k+2,2\ell-k+2,2\ell-k+3),&\ell+2\leq k\leq 2\ell+1.\end{array} \right.\]  
and  
\[C_{R}(y_{2k+1})=\left\{ \begin{array}{ll}(k,k,k-1),&2\leq k\leq\ell;\\ (\ell,\ell+1,\ell),&k=\ell+1;\\ (2\ell-k+1,2\ell-k+2,2\ell-k+2),&\ell+2\leq k\leq 2\ell.\end{array} \right.\]  
Again, all vertices in \(\Gamma\) have a distinct representation, which shows that \(\beta(\Gamma)\leq 3\) when \(n\equiv 2(\mod 4)\). **Subcase 1.3:** When \(n\equiv 3(\mod 4)\). We can write \(n=4\ell+3\), \(\ell\in\mathbb{Z}^{+}\). \[C_{R}(y_{2k})=\left\{