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

08/22/2022

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by   Iain Beaton, et al.

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In this paper we study the the average order of dominating sets in a graph, 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, avd(G) ≤ 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 avd(G) = 2n/3. We also use our bounds to prove the average version of Vizing's Conjecture.