Zhao Wang , Yaping Mao , Sun-Yuan Hsieh , Ralf Klasing , Yuzhi Xiao
{"title":"The g-extra connectivity of graph products","authors":"Zhao Wang , Yaping Mao , Sun-Yuan Hsieh , Ralf Klasing , Yuzhi Xiao","doi":"10.1016/j.jcss.2024.103552","DOIUrl":null,"url":null,"abstract":"<div><p>Connectivity is one of important parameters for the fault tolerant of an interconnection network. In 1996, Fàbrega and Fiol proposed the concept of <em>g</em>-extra connectivity. A subset of vertices <em>S</em> is said to be a <em>cutset</em> if <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> is not connected. A cutset <em>S</em> is called an <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span><em>-cutset</em>, where <em>g</em> is a non-negative integer, if every component of <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> has at least <span><math><mi>g</mi><mo>+</mo><mn>1</mn></math></span> vertices. If <em>G</em> has at least one <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>-cutset, the <em>g-extra connectivity</em> of <em>G</em>, denoted by <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is then defined as the minimum cardinality over all <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>-cutsets of <em>G</em>. In this paper, we first obtain the exact value of <em>g</em>-extra connectivity for the lexicographic product of two general graphs. Next, the upper and lower sharp bounds of <em>g</em>-extra connectivity for the Cartesian product of two general graphs are given. In the end, we apply our results on grid graphs and 2-dimensional generalized hypercubes.</p></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"145 ","pages":"Article 103552"},"PeriodicalIF":1.1000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computer and System Sciences","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022000024000473","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"BUSINESS, FINANCE","Score":null,"Total":0}
引用次数: 0
Abstract
Connectivity is one of important parameters for the fault tolerant of an interconnection network. In 1996, Fàbrega and Fiol proposed the concept of g-extra connectivity. A subset of vertices S is said to be a cutset if is not connected. A cutset S is called an -cutset, where g is a non-negative integer, if every component of has at least vertices. If G has at least one -cutset, the g-extra connectivity of G, denoted by , is then defined as the minimum cardinality over all -cutsets of G. In this paper, we first obtain the exact value of g-extra connectivity for the lexicographic product of two general graphs. Next, the upper and lower sharp bounds of g-extra connectivity for the Cartesian product of two general graphs are given. In the end, we apply our results on grid graphs and 2-dimensional generalized hypercubes.
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