Bounds on the locating-total domination number in trees (Q2282462)

The authors determine lower and upper bounds on the locating-total domination number of trees of order at least three. The locating-total domination number \(\gamma_t^L(G)\) of a graph \(G\) is the minimum cardinality of a locating-total dominating set, that is, a set of vertices that is both a total dominating set (every vertex in the graph has a neighbour in the set) and a locating set (no two vertices outside the set have the same neighbourhood in the set). Two lower bounds and one upper bound are proved in the paper. The upper bound is expressed in terms of \(n\) and \(d\), the order and the diameter of the tree, respectively. The two lower bounds are expressed in terms of the order, the number \(\ell\) of leaves, the number \(s\) of support vertices (neighbours of leaves), and the number \(s_1\) of strong support vertices (vertices adjacent to at least two leaves) of the tree. Extremal trees achieving the given bounds are characterized. In the abstract and in the introduction of the paper, the authors announce that they also show a lower bound in terms of the diameter. However, as they explain in Section 2, this bound holds for all connected graphs of order at least two and was proved by \textit{M. A. Henning} and \textit{N. J. Rad} [Discrete Appl. Math. 160, No. 13--14, 1986--1993 (2012; Zbl 1246.05119)]. Analyzing the proof of this lower bound, the authors characterize extremal trees achieving this bound, too. The four bounds can be summarized as follows: \[ \frac{d+1}{2}\leq \gamma_t^L(T)\leq n-\frac{d-1}{2} \] and \[ \gamma_t^L(T)\geq \max\left\{\frac{n+\ell-s+1}{2}-\frac{s+s_1}{4},\frac{2(n+1)+3(\ell-s)-s_1}{5}\right\}\,. \] The lower bound \(\gamma_t^L(T)\geq \frac{2(n+1)+3(\ell-s)-s_1}{5}\) refines (and implies) the bound \(\gamma_t^L(T)\geq \frac{2(n+\ell-s+1)}{5}\) proved by \textit{M. Chellali} [Discuss. Math., Graph Theory 28, No. 3, 383--392 (2008; Zbl 1173.05034)]. The lower bound \(\gamma_t^L(T)\geq \frac{n+\ell-s+1}{2}-\frac{s+s_1}{4}\) refines (and implies) the bound \(\gamma_t^L(T)\geq \frac{n+\ell+1}{2}-s\) proved by \textit{X.-G. Chen} and \textit{M. Y. Sohn} [Discrete Appl. Math. 159, No. 8, 769--773 (2011; Zbl 1223.05207)].