Many problems of practical interest can be modeled and solved by using fuzzy graph (FG) algorithms. In general, fuzzy graph theory has a wide range of application in various fields. Since indeterminate information is an essential real-life problem and is often uncertain, modeling these problems based on FG is highly demanding for an expert. A vague graph (VG) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems in which fuzzy graphs may not succeed in bringing about satisfactory results. Domination in FGs theory is one of the most widely used concepts in various sciences, including psychology, computer sciences, nervous systems, artificial intelligence, decision-making theory, etc. Many research studies today are trying to find other applications for domination in their field of interest. Hence, in this paper, we introduce different kinds of domination sets, such as the edge dominating set (EDS), the total edge dominating set (TEDS), the global dominating set (GDS), and the restrained dominating set (RDS), in product vague graphs (PVGs) and try to represent the properties of each by giving some examples. The relation between independent edge sets (IESs) and edge covering sets (ECSs) are established. Moreover, we derive the necessary and sufficient conditions for an edge dominating set to be minimal and show when a dominance set can be a global dominance set. Finally, we try to explain the relationship between a restrained dominating set and a restrained independent set with an example. Today, we see that there are still diseases that can only be treated in certain countries because they require a long treatment period with special medical devices. One of these diseases is leukemia, which severely affects the immune system and the body’s defenses, making it impossible for the patient to continue living a normal life. Therefore, in this paper, using a dominating set, we try to categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in the countries better suited in terms of both cost and distance.

Currently, aesthetic education has been integrated into all aspects of talent training. There exists a close correlation between mathematics and beauty, which serves as the practical foundation for implementing mathematics aesthetic education teaching. Mathematical aesthetic education requires teachers to guide students to establish the cognition of mathematical beauty, stimulate students' understanding and appreciation of mathematical beauty, and cultivate students' logical thinking and creativity. However, in actual teaching mathematics teachers lack the theoretical guidance for teaching mathematics aesthetic education, as well as the resources for aesthetic education and the platform for sharing resources. In order to solve the problems facing the teaching of aesthetic education in mathematics, this paper designs a teaching model of aesthetic education in mathematics based on Dynamic Geometry Software. The model is based on the theory of aesthetic psychology and Zhang Dianzhou's theory of the division of the connotation of mathematical beauty, which first divides the process of students' learning of mathematical beauty, and then establishes a teaching process based on this division, including creating environmental beauty, discovering and talking about beauty, appreciating and experiencing beauty, exploring and thinking about beauty, and applying and creating beauty. The incorporation of the NetPad Dynamic Geometry Software allows for the production and sharing of mathematical aesthetic materials, demonstrates the dynamic process of generating mathematical beauty, and allows for a more fluid implementation of teaching models. Under the guidance of this teaching model, the teaching practice of "The rotation of figure" has fully penetrated the thought of mathematical aesthetic education, developed students' aesthetic ability, and achieved good teaching results.