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

TL;DR

The paper advances locating-total domination by establishing the conjectured bound $\gamma_t^L(G)\le \frac{2}{3}n$ for several twin-free graph classes; in some cases it proves stronger bounds, notably $\gamma_t^L(G)\le \frac{n}{2}$ for cobipartite graphs. It develops class-specific proofs: a Bondy-based argument for cobipartite graphs, a direct LTD-set construction for split graphs that yields strict improvement, a block-tree decomposition for block graphs, and a minimal-counterexample plus induction strategy for subcubic graphs. The paper also demonstrates tightness and near-tightness through infinite families such as the $2$-coronas, showing $\gamma_t^L(G)=\frac{2}{3}n$ in some constructions and guiding where the bound is attained. These results provide evidence toward the general conjecture and suggest extensions to chordal and interval graphs and further analysis of cubic 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 $γ^L_t(G)$. It has been conjectured that $γ^L_t(G)\leq\frac{2n}{3}$ 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.

Figures (5)

• Figure 1: The construction of graph $G_k$ in the proof of Proposition \ref{['prop:split-tight']}, with an optimal LTD-set (black vertices).
• Figure 2: Figure (a) represents a twin-free block graph $G$ and Figure (b) represents $T_G$. The vertices underneath the dashed curve represent those deleted from $G$ to obtain $G'$. The black vertices represent vertices in the set $A$. (All notations are as in the proof of Theorem \ref{['theorem_block']}.)
• Figure 3: Twin-free block graph $G$. The dotted boxes mark the blocks $X$, $X"$ and $Y$ of $G$ as in the proof of Theorem \ref{['theorem_block']}.
• Figure 4: The $2$-corona $K_6 \circ P_2$ of a complete graph of order $6$.
• Figure 5: The $2$-corona $C_6 \circ P_2$ of a $6$-cycle.

Theorems & Definitions (23)

• Conjecture 1: conjpaperLTD
• Theorem 3: Bondy B72
• Theorem 4
• proof
• Theorem 5
• proof
• Proposition 6
• proof
• Theorem 7
• proof