Connectivity of directed fuzzy incidence graphs applied to traffic networks

Directed fuzzy incidence graphs can simulate several natural and man-made flow networks with additional flow components. The restriction of flow direction in directed fuzzy incidence graphs enables the modeling of physical problems, such as unauthorized flow through one-way traffic networks and white noise in communication networks, that the fuzzy incidence graph cannot portray adequately. In this article, the idea of trees in directed fuzzy incidence graphs are described and the structure of a legal fuzzy incidence tree is explained using directed incidence connectivity and legal incidence strength. The relationships of legal fuzzy incidence trees to other connectivity ideas, such as fuzzy incidence tournaments, legal flow reduction links, and legal cycles are examined. It is shown that a legal cycle cannot be a legal fuzzy incidence tree and a legal fuzzy incidence tree cannot be a fuzzy incidence tournament. Moreover, the arcs of directed fuzzy incidence graphs are categorized into three, legal a-strong arcs, legal ss-strong arcs, and legal d-arcs. It is proved that an arc in a directed fuzzy incidence graph is legal strong if and only if its legal flow equals the directed incidence connectivity between end nodes. The characterization of legal flow reduction links using legal a-strong arcs is obtained. An arc is a legal flow reduction link if and only if it is legal a-strong. The relation of legal fuzzy incidence trees, and fuzzy incidence tournaments with legal ss-arcs is also identified. As an application, the notion of legal fuzzy incidence trees is applied to a natural model to solve the problem of placing a tollbooth in a suitable location of a one-way traffic system to provide maximum income for the government.