Divisor equitably strong non-split divisor equitable domination in graphs

In epidemiology, the spread of diseases can be modelled using graphs, where individuals are nodes, and edges represent potential pathways for disease transmission. A non-split dominating set could help identify key individuals (or groups) whose monitoring or immunization would ensure that the rest of the population (the non-dominated group) remains connected and can be controlled in case of disease spread. This approach has the potential to have a significant impact across various areas of medicine. We present the idea of non-split divisor equitable domination in graphs as a way to optimize medical networks. Let Q be a graph with vertex set R(Q) and edge set E (Q). Two vertices h and t are known as degree divisor equitable if gcd(dQ(h), dQ(t)) = 1. F ⊂ R(Q) is known as divisor equitable dominating set of Q if ∀ h ∈ R\F, ∋ a t ∈ F such that h and t are adjacent and degree divisor equitable. The divisor equitable domination number of a graph γde(Q) of Q is the minimum cardinality of a divisor equitable dominating set of Q. In this paper, we introduce the concept of a non-split divisor equitable dominating set, divisor equitably strong non-split divisor equitable dominating set, and divisor equitable independent set and divisor equitable clique number. It also explores the concepts of a divisor equitable vertex dominating set, complement divisor equitable graph, and divisor equitable vertex cut.