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arXiv · 2211.14178

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

Dipayan Chakraborty·Florent Foucaud·Anni Hakanen·Michael A. Henning·Annegret K. Wagler
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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.

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Dipayan Chakraborty, Florent Foucaud, Anni Hakanen, Michael A. Henning, Annegret K. Wagler. 2022-11-25. Progress towards the two-thirds conjecture on locating-total dominating sets. https://doi.org/10.1016/j.disc.2024.114176

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