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) be a graph. A set D⊆V is a total restrained dominating set if every vertex in V−D has at least one neighbor in D and at least one neighbor in V−D, and every vertex in D has at least one neighbor in D. The total restrained domination number of G, denoted by γ_tr(G), is the minimum cardinality of all total restrained dominating sets of G. We determine the best possible upper and lower bounds for γ_tr(G), characterize those graphs achieving these bounds and find the best possible lower bounds for where both G and are connected.