Abstract
The class of trees in which the degree of each vertex does not exceed an integer \(d\) is considered. It is shown that, for \(d=4\), each \(n\)-vertex tree in this class contains at most \((\sqrt{2})^n\) minimum dominating sets (MDS), and the structure of trees containing precisely \((\sqrt{2})^n\) MDS is described. On the other hand, for \(d=5\), an \(n\)-vertex tree containing more than \((1/3) \cdot 1.415^n\) MDS is constructed for each \(n \geq 1\). It is shown that each \(n\)-vertex tree contains fewer than \(1.4205^n\) MDS.
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This work was supported by the Russian Science Foundation under grant 21-11-00194.
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 577–595 https://doi.org/10.4213/mzm13571.
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Taletskii, D.S. On the Number of Minimum Dominating Sets in Trees. Math Notes 113, 552–566 (2023). https://doi.org/10.1134/S0001434623030264
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DOI: https://doi.org/10.1134/S0001434623030264