An upper bound on the total restrained domination number of a tree
Let G = (V, E) be a graph. A set of vertices S subset of V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V - S is djacent to a vertex in V - S. The total restrained domination number of G, denoted by gamma(tr)(G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that gamma(tr)(T)  = 3, and s and l are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.
total restrained domination trees