# The Application of Fault-Tolerant Partition Resolvability in Cycle-Related Graphs
## 4 Application  
Consider an example of water flow in a locality through a distribution network system using distribution reservoirs. The whole area is partitioned into a number of distribution sites, where distribution reservoirs are linked to the distribution sites with the help of \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 \(v_{1}\) & 0 & 2 & \(\epsilon+1\) & 1 \\ \hline \(v_{r}(2\leq r\leq\epsilon+2)\) & 0 & 1 & \(\epsilon-r+3\) & \(r\) \\ \hline \(v_{\epsilon+r+1}(2\leq r\leq\epsilon+1)\) & 0 & \(r\) & 1 & \(\epsilon-r+3\) \\ \hline \(v_{2\epsilon+r+2}(2\leq r\leq\epsilon)\) & 0 & \(\epsilon-r+3\) & \(r\) & 1 \\ \hline \(v_{3\epsilon+3}\) & 0 & 2 & \(\epsilon+2\) & 2 \\ \hline \(v_{3\epsilon+r+1}(3\leq r\leq\epsilon+2)\) & 1 & 0 & \(\epsilon-r+4\) & \(r\) \\ \hline \(v_{4\epsilon+r+2}(2\leq r\leq\epsilon+2)\) & 1 & \(r\) & 0 & \(\epsilon-r+4\) \\ \hline \(v_{5\epsilon+r+3}(2\leq r\leq\epsilon+1)\) & 1 & \(\epsilon-r+4\) & \(r\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Distance codes of \(V(f_{a\times 3})\) with respect to \(\aleph\). \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 \(v_{r}(1\leq r\leq\epsilon+1)\) & 0 & 1 & \(\epsilon-r+2\) & \(r\) \\ \hline \(v_{e+r}(2\leq r\leq\epsilon+1)\) & 0 & \(r\) & 1 & \(\epsilon-r+2\) \\ \hline \(v_{2\epsilon+r}(2\leq r\leq\epsilon)\) & 0 & \(\epsilon-r+2\) & \(r\) & 1 \\ \hline \(v_{3\epsilon+r-1}(2\leq r\leq\epsilon+1)\) & 1 & 0 & \(\epsilon-r+3\) & \(r\) \\ \hline \(v_{4\epsilon+r-1}(2\leq r\leq\epsilon+1)\) & 1 & \(r\) & 0 & \(\epsilon-r+3\) \\ \hline \(v_{5\epsilon+r-1}(2\leq r\leq\epsilon+1)\) & 1 & \(\epsilon-r+3\) & \(r\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Distance codes of \(V(f_{a\times 3})\) with respect to \(\aleph\). distribution pipes. The aim is that the system should supply water to all the intended places optimally. The network distribution system could be affected by these factors:  
1 If the reservoirs are located at a far distance from the place demanding water, then more operating power from the pumps should be used. 2 If the number of distribution sites increases, then the distribution time of the water also increases. This scenario can be conveyed in terms of the PD as follows. If in a distribution network system the area is partitioned into a number of distribution sites such that the distribution reservoirs supply water to all intended places, then the minimum number of reservoirs should be located in the minimum number of distribution sites so that each node uniquely recieves or supplies water, depending upon the minimum distance from the node to the partition set, and efficient utility of the network capacity be realized. Consider a distribution network system in the form of \(f_{9\times 3}\), whose nodes are arranged according to the partition sets given in Theorem 2. If one distribution reservoir is located in each of the distribution sites, as shown in Figure 3, then the network permits more flexible operation, and water reaches the given points such that a unique reservoir approaches a unique destination at the minimum distance. This explains the partition dimension of the graph. Furthermore, as the distribution network spreads geographically, and due to managing various network components, there is the possibility of failure of any node in a system. Thus, achieving fault tolerance is vital. Consider a distribution network system whose nodes are arranged according to the partition sets given in Theorem 3. If one-third of the number of vertices in each partition set are distribution reservoirs, as shown in Figure 4, then the service quality of an application will not degrade during the repair of any distribution reservoir of the system, and no consumer will be without a water supply.