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

TL;DR

This paper investigates the two-thirds conjecture for locating-total dominating sets, proving it holds for specific graph classes such as cobipartite, split, block, and subcubic graphs.

Contribution

It confirms the conjecture for several important classes of graphs, advancing understanding of locating-total dominating sets.

Findings

The conjecture holds for cobipartite graphs.

The conjecture holds for split graphs.

The conjecture holds for block and subcubic graphs.

Abstract

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 $\gamma^L_t(G)$. It has been conjectured that $\gamma^L_t(G)\leq\frac{2n}{3}$ 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.