The Average Order of Dominating Sets of a Graph

TL;DR

This paper investigates the average size of dominating sets in graphs, identifying extremal graphs, bounds, and typical behavior, with specific results for trees and graphs without isolated vertices.

Contribution

It determines extremal graphs for maximum and minimum average dominating set size, and provides bounds and asymptotic behavior for various graph classes.

Findings

Star minimizes average dominating set size among trees.

Average order in graphs without isolated vertices is at most 3n/4.

Normalized average tends to 1/2 for almost all graphs.

Abstract

This papers focuses on the average order of dominating sets of a graph. We find the extremal graphs for the maximum and minimum value over all graphs on $n$ vertices, while for trees we prove that the star minimizes the average order of dominating sets. We prove the average order of dominating sets in graphs without isolated vertices is at most $3n/4$, but provide evidence that the actual upper bound is $2n/3$. Finally, we show that the normalized average, while dense in $[1/2,1]$, tends to $\frac{1}{2}$ for almost all graphs.