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arXiv · 2208.10475

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

Iain Beaton·Ben Cameron
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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.

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Iain Beaton, Ben Cameron. 2022-08-22. A Tight Upper Bound on the Average Order of Dominating Sets of a Graph. https://arxiv.org/abs/2208.10475

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