Progress towards the two-thirds conjecture on locating-total dominating sets
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911980980273152 |
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| author | Chakraborty, Dipayan Foucaud, Florent Hakanen, Anni Henning, Michael A. Wagler, Annegret K. |
| author_facet | Chakraborty, Dipayan Foucaud, Florent Hakanen, Anni Henning, Michael A. Wagler, Annegret K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14178 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Progress towards the two-thirds conjecture on locating-total dominating sets Chakraborty, Dipayan Foucaud, Florent Hakanen, Anni Henning, Michael A. Wagler, Annegret K. Combinatorics Discrete Mathematics 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. |
| title | Progress towards the two-thirds conjecture on locating-total dominating sets |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2211.14178 |