Nordhaus-gaddum type results for domination sums in graphs with minimum degree at least four, five or six

Abstract

A vertex set D of a graph G is a dominating set if every vertex not in D is adjacent to some vertex in D. domination number γ(G) of a graph G is the minimum cardinality of a dominating set in G. denote by δ(G) the minimum degree and by G the complement of a graph G. If G is a graph of order n ≠ 10,13,16 such that δ(G), δ(G) > 4 and G,G ≠ K₃ x K₃, then we prove in this paper the Nordhaus-Gaddum type result γ(G) + γ(G) ≤ |4n/ll| + 2, and we investigate the family of graphs achieving this. In addition, we present similar results for graphs G with δ(G),δ(G) > 5 and δ(G),δ(G) > 6.