Progress towards the two-thirds conjecture on locating-total dominating sets

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set S of vertices of a graph G is a locating-total dominating set if every vertex of G has a neighbor in S, and if any two vertices outside S have distinct neighborhoods within S. The smallest size of such a set is denoted by ${\gamma }_t^L\left(G\right)$. It has been conjectured that ${\gamma }_t^L\left(G\right)\le \frac{2n}{3}$ holds for every twin-free graph G of order n without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.