On the average order of a dominating set of a forest

Abstract

We show that the average order of a dominating set of a forest graph G on n vertices with no isolated vertices is at most 2n/3. Moreover, the equality is achieved if and only if every non-leaf vertex of G is a support vertex with one or two leaf neighbors. Our result answers an open question of Beaton and Brown in [The average order of dominating sets of a graph, Discrete Math. 344(12) (2021)]. (C) 2022 Elsevier B.V. All rights reserved.