{"title":"A note on secure domination number in 2K2-free graphs","authors":"Xiaodong Chen, Tianhao Li, Jiayuan Zhang","doi":"10.1016/j.dam.2025.02.019","DOIUrl":null,"url":null,"abstract":"<div><div>A dominating set <span><math><mi>D</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is secure if for each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>D</mi><mo>,</mo></mrow></math></span>\n <span><math><mi>D</mi></math></span> contains a neighbor <span><math><mi>u</mi></math></span> of <span><math><mi>v</mi></math></span> such that <span><math><mrow><mrow><mo>(</mo><mi>D</mi><mo>−</mo><mrow><mo>{</mo><mi>u</mi><mo>}</mo></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>{</mo><mi>v</mi><mo>}</mo></mrow></mrow></math></span> is a dominating set of <span><math><mi>G</mi></math></span>. The minimum cardinality of a secure dominating set in <span><math><mi>G</mi></math></span> is the secure domination number of <span><math><mi>G</mi></math></span> and denoted by <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> A graph is <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free if it does not contain two independent edges as an induced subgraph. Let <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the independence number of <span><math><mrow><mi>G</mi><mo>.</mo></mrow></math></span> Several results gave the upper bound of <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> by a function of <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> In this note, we shows that <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> for every <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph <span><math><mrow><mi>G</mi><mo>;</mo></mrow></math></span> moreover, we give an example to show the bound in our result is best possible.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 162-164"},"PeriodicalIF":1.0000,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X25000939","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A dominating set of a graph is secure if for each vertex
contains a neighbor of such that is a dominating set of . The minimum cardinality of a secure dominating set in is the secure domination number of and denoted by A graph is -free if it does not contain two independent edges as an induced subgraph. Let denote the independence number of Several results gave the upper bound of by a function of In this note, we shows that for every -free graph moreover, we give an example to show the bound in our result is best possible.
期刊介绍:
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