Bounds on the Locating-Total Domination Number in Trees

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 $V$ has a neighbor in $S$. A total dominating set $S$ is called a locating-total dominating set if for each pair of distinct vertices $u$ and $v$ in $V \ S, N(u) ∩ S ≠ N(v) ∩ S$. The minimum cardinality of a locating-total dominating set of $G$ is the locating-total domination number, denoted by $γ_t^L(G)$. We show that, for a tree $T$ of order $n ≥ 3$ and diameter $d$, \(\frac{d+1}{2}≤γ_t^L(T)≤n−\frac{d−1}{2}\), and if $T$ has $l$ leaves, $s$ support vertices and $s_1$ strong support vertices, then \(γ_t^L(T)≥max\Big\{\frac{n+l−s+1}{2}−\frac{s+s_1}{4},\frac{2(n+1)+3(l−s)−s_1}{5}\Big\}\). We also characterize the extremal trees achieving these bounds.