Abstract
Considering all physical, biological, and social systems, fuzzy graph (FG) models serve the elemental processes of all natural and artificial structures. As the indeterminate information is an essential real-life problem, which is mostly uncertain, modeling the problems based on FGs is highly demanding for an expert. Vague graphs (VGs) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed in bringing about satisfactory results. In addition, VGs are a very useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. The previous definition restrictions 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 problems, 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 and 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 article, we introduce different concepts of dominating, equitable dominating, total equitable dominating, weak (strong) equitable dominating, equitable independent, and perfect dominating sets in VGs and also investigate their properties by some examples. Finally, we present an application in medical sciences to show the importance of domination in VGs.
Highlights
Many real-world situations can accessibly be explained by means of a diagram consisting of a set of points together with lines joining certain pairs of these points
A Vague graphs (VGs) is a generalized structure of a fuzzy graph (FG) that provides more exactness, adaptability, and compatibility to a system when matched with systems that run on FGs
A VG is capable of concentrating on determining the uncertainty coupled with the inconsistent and indeterminate information of any realworld problem, where FGs may not lead to adequate results
Summary
Many real-world situations can accessibly be explained by means of a diagram consisting of a set of points together with lines joining certain pairs of these points. In 1965, Zadeh [44] proposed fuzzy set (FS) theory as a model for the exemplification of uncertainty and vagueness in real-world systems. Rashmanlou et al [27,28,29,30,31] advanced new concepts in VGs. A VG is a generalized structure of a FG that provides more exactness, adaptability, and compatibility to a system when matched with systems that run on FGs. In addition, a VG is capable of concentrating on determining the uncertainty coupled with the inconsistent and indeterminate information of any realworld problem, where FGs may not lead to adequate results. Many emergency patients died due to delays in transportation to the hospital; we introduce an application in the transportation system to show the importance of domination in VGs
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