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

Let G=(V,E) be a graph. A set [email protected]?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 [email protected]?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 @c"t"r(G) (@c"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 [email protected]? are not isomorphic to P"3, then 4=<@c"r(G)[email protected]"r([email protected]?)=