On the average order of a dominating set of a forest

TL;DR

This paper proves that in any forest graph with no isolated vertices, the average size of a dominating set is at most two-thirds of the total vertices, with equality characterizing specific support vertex structures.

Contribution

It establishes a tight upper bound on the average dominating set size in forests and characterizes the extremal graphs achieving equality.

Findings

Average dominating set size in forests is at most 2n/3.

Equality holds if and only if each non-leaf is a support vertex with leaf neighbors.

Answers an open question by Beaton and Brown.

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.