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

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Johannes H. Hattingh, Elizabeth Jonck, Ernst J. Joubert, Andrew R. Plummer
• Keywords: DM 2008, Dominating Set, Domination Numbers, Total Restrained Domination