Equitable Domination in Fuzzy Digraphs

Fuzzy digraphs, an extension of directed graphs enhanced with fuzzy set theory, provide a robust framework for representing uncertainty in networks with relationships of varying strengths. In fuzzy digraphs, nodes represent entities, and directed edges are assigned fuzzy values to measure the degree of connection between them. This approach is particularly useful for modeling uncertainty in network applications like communication networks, transportation systems, and social networks, where relationships may be imprecise, uncertain, or dynamic. Equitable domination in a fuzzy graph involves selecting a dominating set where the membership values between dominated and dominating nodes are balanced. The concept of an equitable dominating set is beneficial in situations when a balanced or equitable distribution is required, as it expands on the idea of a traditional dominating set by including a fairness criteria. By providing a sophisticated tool for assessing and creating networks and systems, this idea closes the gap between dominance and equitability. In this article, we extend equitable concept to weighted graphs and apply it in more complex scenarios where vertices have varying levels of importance or weight. We studied the ideas of fuzzy equitable domination in fuzzy digraphs. We defined the fuzzy equitable domination number and studied the properties of minimal fuzzy equitable dominating set and fuzzy equitable independent sets.