Bounds on the differentiating-total domination number of a tree

Given a graph G=(V,E)G=(V,E) with no isolated vertex, a subset SS of VV is called a total dominating set of GG if every vertex in VV is adjacent to a vertex in SS. A total dominating set SS is called a differentiating-total dominating set if for every pair of distinct vertices uu and vv in VV, N[u]∩S≠N[v]∩SN[u]∩S≠N[v]∩S. The minimum cardinality of a differentiating-total dominating set of GG is the differentiating-total domination number of GG, denoted by γtD(G). We show that, for a tree TT of order n≥3n≥3 and diameter dd having ll leaves and ss support vertices, 3(d+1)5≤γtD(T)≤n−2(d−2)5 and 611(n+1+l2−s)≤γtD(T)≤3(n+l)5. Moreover, we characterize the extremal trees achieving these bounds.