On the number of minimum dominating sets and total dominating sets in forests

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.