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

TL;DR

This paper establishes bounds on the maximum number of minimum dominating sets in forests and disproves a conjecture regarding total dominating sets, advancing understanding of dominating set enumeration in trees.

Contribution

It provides new upper bounds for minimum dominating sets and refutes a previous conjecture on total dominating sets in forests.

Findings

Maximum number of minimum dominating sets is at most ^{} in forests.

Constructs trees with many minimum dominating sets exceeding ^{}.

Disproves a conjecture about the number of minimum 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.