Bounds on the differentiating-total domination number of a tree

Given a graphG = (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 D t (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 D t (T ) ≤ n − 2(d−2) 5 and 6 11 (n + 1 + l 2 − s) ≤ γ D t (T ) ≤ 3(n+l) 5 . Moreover, we characterize the extremal trees achieving these bounds. © 2015 Elsevier B.V. All rights reserved.