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On the number of minimum dominating sets and total dominating sets in forests
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abstract
We show that the maximum number of minimum dominating sets of a forest with domination number $\gamma$ is at most $\sqrt{5}^{\gamma}$ and construct for each $\gamma$ a tree with domination number $\gamma$ that has more than $\frac{2}{5}\sqrt{5}^{\gamma}$ minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.
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