Double resolving sets and the exchange property with applications in network optimization and cybersecurity

In the article, we discussed the double resolving sets and exchange property for the octagonal nanosheet structure. We examine the theoretical foundations and practical optimization of double resolving sets in the context of network redundancy and monitoring. A particular focus is placed on the exchange property, which is key in enhancing the flexibility and adaptability of resolving sets under dynamic network conditions. Suppose $W_1$ and $W_2$ are subsets of the vertices of the graph and both have unique identification from all the vertices of the graph. Let $u$ and $v$ are two vertices such that $v\in W_1$ and $u\in W_2.$ It is assumed that $(W_1\setminus v)\cup u=W_2$ then we can say that exchange property is hold. The concept of a double resolving set ensures that every vertex in a graph can be uniquely identified using two independent resolving sets, providing the exchange property in cases where one vertex becomes ineffective. By applying sophisticated mathematical methods, we determine dual resolving sets that offer the best vertex identification and distance resolution in carbon nanosheets. These sets are essential for use in electronic devices and sensors, for example, where exact control over the characteristics of nanosheets is required. We also investigate the increased exchange qualities, which are these sets’ resilience to external shocks and their capacity to continue addressing problems even when flaws or alterations are present. Suppose $W_1$ and $W_2$ resolving sets and $W_1$ $\ne$ $W_2$ then $G$ has double resolving set.