Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number γ has at most 2 γ minimum dominating sets. Bień gave a counterexample, which allows to construct forests with domination number γ and 2.0598 γ minimum dominating sets. We show that every forest with domination number γ has at most 2.4606 γ minimum dominating sets, and that every tree with independence number α has at most 2 α−1 + 1 maximum independent sets.

No abstract

No abstract

We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating function. Our characterization is based on five simple extension operations, and reveals several structural properties of these trees.

No abstract