Geodesic
On the number of minimum total dominating~sets~in~trees
Diskretnyj analiz i issledovanie operacij, Tome 30 (2023) no. 1, pp. 110-129

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The minimum total dominating set (MTDS) of a graph is a vertex subset $D$ of minimum cardinality such that every vertex of the graph is adjacent to at least one vertex of $D.$ In this paper we obtain the sharp upper bound for the number of MTDS in the class of $n$-vertex $2$-caterpillars. We also show that for all $n \geq 1$ every $n$-vertex tree has less than $(\sqrt{2})^n$ MTDS. Illustr. 5, bibliogr. 6.
Keywords: extremal combinatorics, tree, $2$-caterpillar, minimum total dominating set. 
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     author = {D. S. Taletskii},
     title = {On the number of minimum total dominating~sets~in~trees},
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D. S. Taletskii. On the number of minimum total dominating~sets~in~trees. Diskretnyj analiz i issledovanie operacij, Tome 30 (2023) no. 1, pp. 110-129. http://geodesic.mathdoc.fr/item/DA_2023_30_1_a5/