Pythagorean fuzzy incidence graphs with application in one-way toll road network

Incidence graphs are an effective to model interconnected networks with additional vertex-edge interactions. They are widely used to establish modes of operation and controllers to illustrate the influence of one factor on another. The purpose of this paper is to present the concept of directed Pythagorean fuzzy incidence graphs. Physical problems that Pythagorean fuzzy incidence graphs cannot effectively illustrate can be modeled by restricting the flow direction because the interactions in these graphs are not symmetric. We discuss the connectivity of directed Pythagorean fuzzy incidence graphs differently by emphasizing legal and illegal network flows. Furthermore, we introduce the concept of legal and illegal flow reduction vertices, edges, and pairs in directed Pythagorean fuzzy incidence graphs and give some significant results. We study the types of legal edges in directed Pythagorean fuzzy incidence graphs and establish some results about legal strong edges. Moreover, we discuss the application of legal fuzzy incidence trees in decision-making, that is, to select the optimal location of the electronic toll collection system on one-way toll roads to maximize toll revenue. Additionally, we provide an algorithm to understand the methods we use in our application. Finally, we compare the proposed method with an existing model that includes numerical results and logical arguments to demonstrate its feasibility and applicability.