On the maximum number of minimum dominating sets in forests

TL;DR

This paper investigates bounds on the number of minimum dominating and maximum independent sets in forests and trees, providing improved exponential upper bounds and addressing a previously posed question.

Contribution

It establishes new upper bounds of $2.4606^\gamma$ for minimum dominating sets in forests and $2^{\alpha-1}+1$ for maximum independent sets in trees, advancing understanding of combinatorial limits.

Findings

Forests with domination number $\gamma$ have at most $2.4606^\gamma$ minimum dominating sets.

Trees with independence number $\alpha$ have at most $2^{\alpha-1}+1$ maximum independent sets.

Improves previous bounds and answers open questions in graph theory.

Abstract

Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $\gamma$ has at most $2^\gamma$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $\gamma$ and $2.0598^\gamma$ minimum dominating sets. We show that every forest with domination number $\gamma$ has at most $2.4606^\gamma$ minimum dominating sets, and that every tree with independence number $\alpha$ has at most $2^{\alpha-1}+1$ maximum independent sets.