Bipolar fuzzy graph is more precise than a fuzzy graph when dealing with imprecision as it is focusing on the positive and negative information of each vertex and edge. Nowadays, researchers have utilized bipolar fuzzy graphs in decision-making problems. Bipolar fuzzy competition graphs aid to compute the competition between the vertices in bipolar fuzzy graphs. To depict the best competitions among the competitions of bipolar fuzzy graphs, the best bipolar fuzzy competition graph can be defined using bipolar fuzzy α-cut and the strength of the competition between the vertices can also be determined. Fuzzy graphs are used well to frame modelling in real-time problems. In particular, when the real-time scenario is modelled using the bipolar fuzzy graph, it gives more precision and flexibility. At present, researchers have focused on decision-making techniques with bipolar fuzzy graphs. The DEMATEL method is one of the powerful decision-making tools. It effectively analyses the complicated digraphs and matrices. The fuzzy DEMATEL technique can convert the interrelations between factors into an intelligible structural model of the system and divide them into cause and effect groups. Therefore, this study attempts to design the DEMATEL method under the bipolar fuzzy environment. To illustrate this proposed technique, the problem of identifying the best mobile network is taken. With this method, the benefits and drawbacks of networks are measured and a complicated bipolar fuzzy directed graph can be transformed into a viewed structure.

The present study aims to introduce a bipolar intuitionistic fuzzy competition graph which is the generalization of competition graphs. In a bipolar intuitionistic fuzzy digraph, the targets are valuable as it depends on the competition power of the competitors. Also, the number of competitors may vary sometimes. Based on these clues, this study presents two generalizations of bipolar intuitionistic fuzzy competition graphs namely bipolar intuitionistic fuzzy k-competition graph and bipolar intuitionistic fuzzy p-competition graph. The concept of the directed bipolar intuitionistic fuzzy graph is coined in this study. Moreover, some results on the bipolar intuitionistic fuzzy graph are derived.

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