# The Application of Fault-Tolerant Partition Resolvability in Cycle-Related Graphs
## 1 Introduction and Preliminaries
: _Let \(Y\) be a graph of the order \(m\geq 5\). If \(Y\) has a node of a degree of at least \(4\), then \(\mathcal{P}(Y)\geq 4\)._  
The present paper is framed in this manner. Section 2 is concerned with the computation of the exact FTPD of a kayak paddle graph, and Section 3 concerns computing the FTPD of a class of flower graphs. Section 4 includes the application of an FTPD in a scenario of installing water reservoirs in a locality. Finally, in the concluding section, we close the paper by giving anticipatory remarks about future work. **2. FTPD of a Kayak Paddle Graph**  
We compute the FTPD of a Kayak paddle graph in this section. The kayak paddle graph, denoted by \(KP(l,t,s)\), has two cycles of lengths \(l\) and \(t\) connected by a path of a length \(s\). The vertex set and edge set of the kayak paddle graph for \(l\geq 3\), \(t\geq 3\) and \(s\geq 2\), are \(V(KP(l,t,s))=\{v_{1},v_{2},\ldots,v_{l+l+s-1}\}\) and \(E(KP(l,t,s))=\{v_{l}v_{l+1}:1\leq j\leq l-1\}\cup\{v_{1}v_{l}\}\cup\{v_{l}v_{l+ 1}:l+1\leq j\leq l+t-1\}\cup\{v_{l+1}v_{l+t}\}\cup\{v_{1}v_{l+t+1}\}\cup\{v_{ l}v_{l+1}:l+t+1\leq j\leq l+t+s-2\}\cup\{v_{l+t+s-1}v_{l+1}\}\), respectively. The kayak paddle graph \(KP(9,7,6)\) is shown in Figure 1.  
The \(\beta(KP(l,t,s))\) was computed in [11]:  
**Lemma 2** ([11]). : _For \(l\geq 3\), \(t\geq 3\) and \(s\geq 2\), \(\beta(KP(l,t,s))=2\)._  
The \(pd(KP(l,t,s))\) was computed in [17]:  
**Lemma 3** ([17]). : _For \(l\geq 3\), \(t\geq 3\) and \(s\geq 2\), \(pd(KP(l,t,s))=3\)._  
The ensuing theorem will allow us to compute \(\mathcal{P}(KP(l,t,s))\) of the kayak paddle graph:  
**Theorem 1**. : _For \(l\geq 3\), \(t\geq 3\) and \(s\geq 2\), \(\mathcal{P}(KP(l,t,s))=4\)._  
Figure 1: Kayak paddle graph \(KP(9,7,6)\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Distance of Vertex** & **From \(\aleph_{1}\)** & **From \(\aleph_{2}\)** & **From \(\aleph_{3}\)** & **From \(\aleph_{4}\)** \\ \hline \(v_{r}(1\leq r\leq\epsilon)\) & 0 & \(r\) & \(r\) & \(s+r\) \\ \hline \(v_{\epsilon+1}\) & 0 & \(\epsilon-1\) & \(\epsilon\) & \(s+\epsilon+1\) \\ \hline \(v_{2\epsilon-r}(1\leq r\leq\epsilon-2)\) & 0 & \(r\) & \(r+1\) & \(s+r+3\) \\ \hline \(v_{2\epsilon}\) & 1 & 0 & 1 & \(s+3\) \\ \hline \(v_{2\epsilon+1}\) & 1 & 1 & 0 & \(s+2\) \\ \hline \(v_{2\epsilon+2}\) & 0 & 1 & \(s+1\) & 1 \\ \hline \(v_{l+t-1}\) & 1 & 0 & \(s+2\) & 1 \\ \hline \(v_{l+t}\) & 1 & 1 & \(s+2\) & 0 \\ \hline \(v_{l+t+r}(1\leq r\leq\lfloor\frac{s}{2}\rfloor)\) & \(r\) & 0 & \(r+1\) & \(s-r+1\) \\ \hline \(v_{l+t+s-r}(1\leq r\leq\lceil\frac{s}{2}\rceil-1)\) & \(r\) & 0 & \(s-r+1\) & \(r+1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distance codes of \(V(KP(l,t,s))\) with respect to \(\aleph\) for \(l=2\epsilon+1\), where \(\epsilon\geq 2\), and \(t=2b+1\), where \(b\geq 2\). It is apparent from Tables 1-6 that \(\aleph\) is a fault-tolerant resolving partition set of \(KP(l,t,s)\), and therefore, \(\mathcal{P}(KP(l,t,s))\leq 4\). Now, we establish that \(\mathcal{P}(KP(l,t,s))\geq 4\). For this, we exhibit that \(\mathcal{P}(KP(l,t,s))\
eq 3\). For a contradiction, we estimate that \(\aleph=\{\aleph_{1},\aleph_{2},\aleph_{3}\}\) is a fault-tolerant partition basis of \(KP(l,t,s)\). As \(v_{1}\) and \(v_{l+1}\) are only vertices of a degree of three, without loss of generality, we assume that \(v_{1}\in\aleph_{1}\) and \(N(v_{1})=\{v_{2},v_{l},v_{l+t+1}\}\). Suppose that \(|\aleph_{1}|=1\) and \(N(v_{1})\subseteq\aleph_{2}\cup\aleph_{3}\), \(|N(v_{1})\cap\aleph_{2}|\geq 2\) or \(|N(v_{1})\cap\aleph_{3}|\geq 2\). Without loss of generality, we suppose that at least two vertices \(a,b\in N(v_{1})\cap\aleph_{2}\). As \(r(a|\aleph)=(1,0,c_{1})\) and \(r(b|\aleph)=(1,0,c_{2})\) are the same in two places, this is hence a contradiction.