On locating and differentiating-total domination in trees

A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let ${\displaystyle \gamma {ₜ}^L\left(G\right)}$ and ${\displaystyle \gamma {ₜ}^D\left(G\right)}$ be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with l leaves and s support vertices, ${\displaystyle \gamma {ₜ}^L\left(T\right)\ge max\{2\frac{n+l-s+1}{5},\frac{n+2-s}{2}\}}$, and for a tree of order n ≥ 3, ${\displaystyle \gamma {ₜ}^D\left(T\right)\ge 3\frac{n+l-s+1}{7}}$, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying ${\displaystyle \gamma {ₜ}^L\left(T\right)=2\frac{n+l-s+1}{5}}$ or ${\displaystyle \gamma {ₜ}^D\left(T\right)=3\frac{n+l-s+1}{7}}$.