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 γtr(G) (γr(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 [3]) that if G is a graph of order n ≥ 2 such that both G and G are not isomorphic to P3, then 4 ≤ γr(G)+γr(G) ≤ n+2. We also provide characterizations of the extremal graphs G of order n achieving these bounds.