A Tight Upper Bound on the Average Order of Dominating Sets of a Graph

TL;DR

This paper proves a conjecture that the average order of dominating sets in any graph without isolated vertices is at most two-thirds of the number of vertices, and characterizes the extremal graphs.

Contribution

It confirms the conjecture by Beaton and Brown, classifies extremal graphs, and applies the bounds to the average version of Vizing's Conjecture.

Findings

Proved the conjecture that vd(G)   2n/3 for all graphs without isolated vertices.

Characterized the graphs achieving the bound vd(G) = 2n/3.

Applied the bounds to establish the average version of Vizing's Conjecture.

Abstract

In this paper we study the the average order of dominating sets in a graph, $\operatorname{avd}(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown (2021) conjectured that for all graphs $G$ of order $n$ without isolated vertices, $\operatorname{avd}(G) \leq 2n/3$. Recently, Erey (2021) proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $\operatorname{avd}(G) = 2n/3$. We also use our bounds to prove the average version of Vizing's Conjecture.