Progress towards the two-thirds conjecture on locating-total dominating sets - Archive ouverte HAL
Article Dans Une Revue Discrete Mathematics Année : 2024

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

Résumé

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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hal-03876474 , version 1 (28-11-2022)
hal-03876474 , version 2 (07-08-2024)

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Dipayan Chakraborty, Florent Foucaud, Anni Hakanen, Michael Henning, Annegret Wagler. Progress towards the two-thirds conjecture on locating-total dominating sets. Discrete Mathematics, 2024, 347 (12), pp.114176. ⟨10.1016/j.disc.2024.114176⟩. ⟨hal-03876474v2⟩
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