Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs

Let G = (V, E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V - S is adjacent to a vertex in V - S. A set S ⊆ V is a restrained dominating set if every vertex in V - S is adjacent to a vertex in S and to a vertex in V - S. The total restrained domination number of G (restrained domination number of G, respectively), denoted by γ_tr (G) (γ_r (G), respectively), is the smallest cardinality of a total restrained dominating set (restrained dominating set, respectively) of G. We bound the sum of the total restrained domination numbers of a graph and its complement, and provide characterizations of the extremal graphs achieving these bounds. It is known (see [G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, L.R. Markus, Restrained domination in graphs, Discrete Math. 203 (1999) 61-69.]) that if G is a graph of order n ≥ 2 such that both G and over(G, -) are not isomorphic to P₃, then 4 ≤ γ_r (G) + γ_r (over(G, -)) ≤ n + 2. We also provide characterizations of the extremal graphs G of order n achieving these bounds.