A tight upper bound on the average order of dominating sets of a graph

In this paper we study the average order of dominating sets in a graph, avd ( G ) $\,\text{avd}\,(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs G $G$ of order n $n$ without isolated vertices, avd ( G ) ≤ 2 n / 3 $\,\te...

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Bibliographic Details
Published inJournal of graph theory Vol. 107; no. 3; pp. 463 - 477
Main Authors Beaton, Iain, Cameron, Ben
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.11.2024
Subjects
Apexes
average graph parameters
dominating set
domination polynomial
Graph theory
Graphs
Upper bounds
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ISSN0364-9024
1097-0118
DOI10.1002/jgt.23143

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Summary:In this paper we study the average order of dominating sets in a graph, avd ( G ) $\,\text{avd}\,(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs G $G$ of order n $n$ without isolated vertices, avd ( G ) ≤ 2 n / 3 $\,\text{avd}\,(G)\le 2n/3$. Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have avd ( G ) = 2 n / 3 $\,\text{avd}\,(G)=2n/3$. We also use our bounds to prove an average version of Vizing's conjecture.
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ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.23143