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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| Published in | Journal of graph theory Vol. 107; no. 3; pp. 463 - 477 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.11.2024
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| ISSN | 0364-9024 1097-0118 |
| DOI | 10.1002/jgt.23143 |
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| Abstract | 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. |
|---|---|
| AbstractList | In this paper we study the average order of dominating sets in a graph, . Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs of order without isolated vertices, . Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have . We also use our bounds to prove an average version of Vizing's conjecture. 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. 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)≤2n/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)=2n/3 $\,\text{avd}\,(G)=2n/3$. We also use our bounds to prove an average version of Vizing's conjecture. |
| Author | Cameron, Ben Beaton, Iain |
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| Cites_doi | 10.1007/s00283-012-9275-2 10.1002/jgt.22434 10.1016/0095-8956(83)90049-7 10.1002/jgt.22793 10.1016/j.ejc.2021.103388 10.1090/proc/13728 10.1016/j.disc.2022.112799 10.1016/0095-8956(84)90046-7 10.1214/aoms/1177703287 10.1016/j.disc.2022.113127 10.1016/j.jctb.2009.05.006 10.1007/s00373-013-1306-z 10.1016/0012-365X(77)90144-3 10.1112/jlms.12056 10.1002/jgt.22621 10.1016/j.dam.2015.12.010 10.1016/B978-1-898563-76-1.50006-8 10.1021/ja01193a005 10.1002/jgt.22359 10.1007/s00373-020-02136-1 10.1016/j.jctb.2021.09.008 10.1007/s40879-019-00333-8 10.1016/j.disc.2021.112595 |
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| Snippet | 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... In this paper we study the average order of dominating sets in a graph, . Like other average graph parameters, the extremal graphs are of interest. Beaton and... 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... |
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| SubjectTerms | Apexes average graph parameters dominating set domination polynomial Graph theory Graphs Upper bounds |
| Title | A tight upper bound on the average order of dominating sets of a graph |
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