The Average Order of Dominating Sets of a Graph

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 1 2 for almost all graphs.

On the average order of a dominating set of a forest

“…Our focus in this paper will be on the average order of a dominating set of a graph. A subset of vertices S is called a dominating set of G if every vertex in V (G) \ S is adjacent to some vertex in S. Let D(G) denote the family of all dominating sets of G. Recently Beaton and Brown [4] introduced the average order of a dominating set of G, denoted avd(G), which is given by…”

“…They showed that the complete graph K n uniquely minimizes the average order of a dominating set among all graphs on n vertices [4]. It is trivial that…”

“…Which graphs have the largest average order of a dominating set among all graphs of order n without isolated vertices? While this question was examined in [4], the question remained unanswered in general. It was shown that avd(G) ≤ 3 4 n for every graph G of order n without isolated vertices [4].…”

“…While this question was examined in [4], the question remained unanswered in general. It was shown that avd(G) ≤ 3 4 n for every graph G of order n without isolated vertices [4]. However the factor 3/4 in the upper bound is not best possible, and the evidence provided in [4] suggests that the constant 3/4 can be improved to a smaller number.…”

“…It was shown that avd(G) ≤ 3 4 n for every graph G of order n without isolated vertices [4]. However the factor 3/4 in the upper bound is not best possible, and the evidence provided in [4] suggests that the constant 3/4 can be improved to a smaller number. Indeed, the following conjecture was proposed.…”