Bounds on the differentiating-total domination number of a tree

Given a graph G = ( V , E ) with no isolated vertex, a subset S of V is called a total dominating set of G if every vertex in V is adjacent to a vertex in S . A total dominating set S is called a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N u ? S ? N v ? S . The minimum cardinality of a differentiating-total dominating set of G is the differentiating-total domination number of G , denoted by γ t D ( G ) . We show that, for a tree T of order n ? 3 and diameter d having l leaves and s support vertices, 3 ( d + 1 ) 5 ? γ t D ( T ) ? n - 2 ( d - 2 ) 5 and 6 11 ( n + 1 + l 2 - s ) ? γ t D ( T ) ? 3 ( n + l ) 5 . Moreover, we characterize the extremal trees achieving these bounds.