Optimizing Decision-Making Using Domination Theory in Product Bipolar Fuzzy Graphs

The bipolar fuzzy model is a rapidly evolving research area that provides a robust framework for addressing real-world problems, with wide-ranging applications in scientific and technical domains. Within this framework, bipolar fuzzy graphs play a significant role in decision-making and problem-solving, particularly through domination theory, which helps tackle practical challenges. This study explores various operations on product bipolar fuzzy graphs, including union (∪), join (+), intersection (∩), Cartesian product (×), composition (∘), and complement, leading to the generation of new graph structures. Several important results related to complete product bipolar fuzzy graphs under these operations are established. Additionally, we introduce key concepts such as dominating sets, minimal dominating sets, and the domination number $⋎\left(\mathcal{H}\right)$, supported by illustrative examples. This study further investigates the properties of domination in the context of these operations. To demonstrate practical applicability, we present a decision-making problem involving the optimization of bus routes and the strategic placement of bus stations using domination principles. This research contributes to the advancement of bipolar fuzzy graph theory and its practical applications in real-world scenarios.