Domination in Exact Deg-Centric Graphs

The exact deg-centric graph associated with a simple, connected graph G, denoted by $$G_{ed}$$ , is defined by $$V(G_{ed}) = V(G)$$ and $$E(G_{ed}) = \{v_i v_j: d_G(v_i, v_j) = \deg _G(v_i)\}$$ . In this paper, we undertake a systematic investigation of domination in exact deg-centric graphs, with particular emphasis on the characterisation of dominating sets and the determination of the domination number. We establish several fundamental results that reveal the interplay between distance-based vertex interactions and degree constraints inherent in the construction of $$G_{ed}$$ . Furthermore, we analyse key structural properties of exact deg-centric graphs and examine their implications for domination parameters. To illustrate the theoretical findings, we derive explicit results for important classes of graphs, including paths, cycles, and complete graphs. These results provide deeper insight into the behaviour of domination in exact deg-centric structures and suggest directions for further research in distance degree-based graph transformations.