Certain Concepts in Directed Rough Fuzzy Graphs and Application to Mergers of Companies

A directed rough fuzzy graph (DRFG) is a unique and innovative hybrid model because it deals with more complex problems of uncertainty in the presence of incomplete data information or rough universe. A DRFG can be obtained from two given DRFGs by union, Cartesian product and composition. When we study operations for DRFGs with a large number of vertices, the degree of vertices in a DRFG presents a confusing picture. Therefore, a mechanism for determining the degree of vertices for DRFG operations is needed. The main objective of this study is to analyze and investigate the degree of vertices in DRFGs formed by certain operations, which will provide clear explanations of operations on DRFGs and their effects on vertex degrees with examples. In this paper, we find the degree of a vertex in DRFGs formed by these operations in terms of the degree of vertices in the given DRFGs in some special cases. We explain these operations with some examples. In addition, we provide an application to the corporate merger problem to test our approach and obtain an optimal result. We have developed two algorithms to elaborate the procedure for our application. Finally, we created a comparison table comparing our results for Algorithms 1 and 2 for the same enterprise merger network.