{"title":"Optimal linear-Vizing relationships for (total) domination in graphs","authors":"Michael A. Henning, Paul Horn","doi":"10.1002/jgt.23070","DOIUrl":null,"url":null,"abstract":"<p>A total dominating set in a graph <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is a set of vertices of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> such that every vertex is adjacent to a vertex of the set. The total domination number <math>\n <semantics>\n <mrow>\n <msub>\n <mi>γ</mi>\n \n <mi>t</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${\\gamma }_{t}(G)$</annotation>\n </semantics></math> is the minimum cardinality of a total dominating set in <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. In this paper, we study the following open problem posed by Yeo. For each <math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n <annotation> ${\\rm{\\Delta }}\\ge 3$</annotation>\n </semantics></math>, find the smallest value, <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n </mrow>\n <annotation> ${r}_{{\\rm{\\Delta }}}$</annotation>\n </semantics></math>, such that every connected graph <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> of order at least 3, of order <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>, size <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n </mrow>\n <annotation> $m$</annotation>\n </semantics></math>, total domination number <math>\n <semantics>\n <mrow>\n <msub>\n <mi>γ</mi>\n \n <mi>t</mi>\n </msub>\n </mrow>\n <annotation> ${\\gamma }_{t}$</annotation>\n </semantics></math>, and bounded maximum degree <math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n </mrow>\n <annotation> ${\\rm{\\Delta }}$</annotation>\n </semantics></math>, satisfies <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n \n <mo>≤</mo>\n \n <mfrac>\n <mn>1</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>Δ</mi>\n \n <mo>+</mo>\n \n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>−</mo>\n \n <msub>\n <mi>γ</mi>\n \n <mi>t</mi>\n </msub>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $m\\le \\frac{1}{2}({\\rm{\\Delta }}+{r}_{{\\rm{\\Delta }}})(n-{\\gamma }_{t})$</annotation>\n </semantics></math>. Henning showed that <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n \n <mo>≤</mo>\n \n <mi>Δ</mi>\n </mrow>\n <annotation> ${r}_{{\\rm{\\Delta }}}\\le {\\rm{\\Delta }}$</annotation>\n </semantics></math> for all <math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n <annotation> ${\\rm{\\Delta }}\\ge 3$</annotation>\n </semantics></math>. Yeo significantly improved this result and showed that <math>\n <semantics>\n <mrow>\n <mn>0.1</mn>\n \n <mi>ln</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>Δ</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo><</mo>\n \n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n \n <msqrt>\n <mi>Δ</mi>\n </msqrt>\n </mrow>\n <annotation> $0.1\\mathrm{ln}({\\rm{\\Delta }})\\lt {r}_{{\\rm{\\Delta }}}\\le 2\\sqrt{{\\rm{\\Delta }}}$</annotation>\n </semantics></math> for all <math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n <annotation> ${\\rm{\\Delta }}\\ge 3$</annotation>\n </semantics></math>, and posed as an open problem to determine “whether <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n </mrow>\n <annotation> ${r}_{{\\rm{\\Delta }}}$</annotation>\n </semantics></math> grows proportionally with <math>\n <semantics>\n <mrow>\n <mi>ln</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>Δ</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\mathrm{ln}({\\rm{\\Delta }})$</annotation>\n </semantics></math> or <math>\n <semantics>\n <mrow>\n <msqrt>\n <mi>Δ</mi>\n </msqrt>\n </mrow>\n <annotation> $\\sqrt{{\\rm{\\Delta }}}$</annotation>\n </semantics></math> or some completely different function.” In this paper, we determine the growth of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n </mrow>\n <annotation> ${r}_{{\\rm{\\Delta }}}$</annotation>\n </semantics></math>, and show that <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mi>Δ</mi>\n </msub>\n </mrow>\n <annotation> ${r}_{{\\rm{\\Delta }}}$</annotation>\n </semantics></math> is asymptotically <math>\n <semantics>\n <mrow>\n <mi>ln</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>Δ</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\mathrm{ln}({\\rm{\\Delta }})$</annotation>\n </semantics></math> and likewise determine the asymptotics of the analogous constant for standard domination.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 1","pages":"149-166"},"PeriodicalIF":0.9000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23070","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A total dominating set in a graph is a set of vertices of such that every vertex is adjacent to a vertex of the set. The total domination number is the minimum cardinality of a total dominating set in . In this paper, we study the following open problem posed by Yeo. For each , find the smallest value, , such that every connected graph of order at least 3, of order , size , total domination number , and bounded maximum degree , satisfies . Henning showed that for all . Yeo significantly improved this result and showed that for all , and posed as an open problem to determine “whether grows proportionally with or or some completely different function.” In this paper, we determine the growth of , and show that is asymptotically and likewise determine the asymptotics of the analogous constant for standard domination.
期刊介绍:
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
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