We introduce and investigate the “fair domination subdivision number” of a graph, a new parameter that combines the concepts of fair domination and edge subdivision. A fair dominating set of a graph His a dominating setJ is a subset of V such that every vertex in V∖J is adjacent to the same number of vertices in J. The “fair domination number”, denotedγ_fd (H), is the minimum cardinality of such a set. The fair domination subdivision number, denoted 〖Sd〗_(γ_fd)^+ (H) (or 〖Sd〗_(γ_fd)^- (H)) is defined as the minimum number of edges in H that must be subdivided each exactly once in order to increase (or decrease) the fair domination number of the resulting graph. We explore the structural implications of edge subdivision on fair domination and establish bounds for 〖Sd〗_(γ_fd)^+ (H) (or〖 Sd〗_(γ_fd)^- (H)) for various classes of graphs. The study shows how minimal subdivisions affect fair domination, revealing stability and sensitivity in domination structures. Characterizations and open problems are discussed to inspire further research. It proposes extending the concept beyond trees, developing efficient and scalable algorithms, and exploring approximation methods. Further research may uncover theoretical properties and connections with other graph parameters, enriching domination theory’s applications.
