Total restrained domination in graphs

In this paper, we initiate the study of a variation of standard domination, namely total restrained domination. Let G=(V,E)G=(V,E) be a graph. A set D⊆VD⊆V is a total restrained dominating set   if every vertex in V−DV−D has at least one neighbor in DD and at least one neighbor in V−DV−D, and every vertex in DD has at least one neighbor in DD. The total restrained domination number   of GG, denoted by γtr(G)γtr(G), is the minimum cardinality of all total restrained dominating sets of GG. We determine the best possible upper and lower bounds for γtr(G)γtr(G), characterize those graphs achieving these bounds and find the best possible lower bounds for γtr(G)+γtr(Ḡ) where both GG and Ḡ are connected.