On the maximum number of minimum dominating sets in forests

J. D. Alvarado, S. Dantas, E. Mohr, D. Rautenbach

Published 2018-03-31, updated 2018-04-11Version 2

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.