This study examines the δ-complements of graphs—a specific type of graph complement whose adjacency depends on the adjacency of the vertices with identical degrees in the original graph. More specifically, we study this type of complement regarding the domination number. We provide sharp Nordhaus–Gaddum-type bounds on the domination number of a graph and its δ-complement. We also provide sharp bounds on the domination numbers of the δ-complements of joined graphs and Cartesian product graphs.
Tangjai et al. (Sun,) studied this question.