Equitable Domination in Vague Graphs With Application in Medical Sciences

Fuzzy graph (FG) models are the building blocks of all natural and artificial structures, specifically dynamic process in physical, biological and social systems. Consistent with the indeterminate information being an integral section of real-life problems, which are mostly uncertain, modeling those problems based on FG is highly demanding for an expert. VG can manage the uncertainty associated with the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed into bringing about satisfactory results. Also, VGs are so useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. The previous definitions limitations in FGs have made us present new definitions in VGs. A wide range of applications has been attributed to the domination in graph theory for several fields such as facility location problem, school bus routing, modeling biological networks, and coding theory. Concepts from domination also exist in problems involving finding the set of representatives, in monitoring communication, electrical networks, and in land surveying (e.g., minimizing the number of places a surveyor must stand in order to take the height measurement for an entire region). Hence, in this paper, we introduce different concepts of dominating, equitable dominating, total equitable dominating, weak (strong) equitable dominating, equitable independent sets, and perfect dominating set in VGs, and also investigate their properties by some examples. Finally, we represent an application in medical sciences to show the importance of domination in VGs.