Applying disconnected domination on well-known graphs and complement graphs

In this paper, a new domination concept in graph theory, referred to as doubly disconnected domination, is introduced. Let be an undirected, nontrivial, finite and simple graph. A subset is called a doubly disconnected dominating set if it is a dominating set, which is meaning that any vertex in has at least one neighbor in the set , and both the induced subgraphs and are disconnected. The least number of elements of such a set among all possible doubly disconnected dominating sets in is called the parameter known as the doubly disconnected domination number. This paper aims to establish several relations for , as well as to analyze its behavior in various graph structures. Additionally, we explore the relationship between and some complement graphs, deriving specific results that determine how this domination parameter transforms under graph complementation. Furthermore, explicit evaluations of are provided for well-known graphs, and certain classes of graphs are identified that do not admit such a domination structure. The results contribute to a deeper understanding of domination properties in graph theory and open new avenues for further exploration in structural and combinatorial graph analysis.