A Study on Regular Domination in Vague Graphs with Application

Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In graph theory, a dominating set (DS) for a graph $G^{\ast }=\left(X,E\right)$ is a subset $\mathfrak{S}$ of the vertices $X$ such that every vertex not in $\mathfrak{S}$ is adjacent to at least one member of $\mathfrak{S}$ . The concept of DS in FGs has received the attention of many researchers due to its many applications in various fields such as computer science and electronic networks. In this paper, we introduce the notion of $\left(\left({ϵ}_1,{ϵ}_2\right),2\right)$ -Regular vague dominating set and provide some examples to explain various concepts introduced. Also, some results were discussed. Additionally, the $\left(\left({ϵ}_1,{ϵ}_2\right),2\right)$ -Regular strong (weak) and independent strong (weak) domination sets for vague domination set (VDS) were presented with some theorems to support the context.