Nordhaus–Gaddum results for restrained domination and total restrained domination in graphs Original Research Article

Let image be a graph. A set image is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of image is adjacent to a vertex in image. A set image is a restrained dominating set if every vertex in image is adjacent to a vertex in S and to a vertex in image. The total restrained domination number of G (restrained domination number of G, respectively), denoted by image (image, 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 image such that both G and image are not isomorphic to image, then image. We also provide characterizations of the extremal graphs G of order n achieving these bounds.