Muhammad Shoaib Sardar, Hamna Choudhry, Jia-Bao Liu
{"title":"Locating Edge Domination Number of Some Classes of Claw-Free Cubic Graphs","authors":"Muhammad Shoaib Sardar, Hamna Choudhry, Jia-Bao Liu","doi":"10.1155/2024/1182858","DOIUrl":null,"url":null,"abstract":"Let <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 20.155 11.5564\" width=\"20.155pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.524,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"23.7371838 -9.28833 14.99 11.5564\" width=\"14.99pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,23.787,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,28.285,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,35.813,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"40.9061838 -9.28833 12.769 11.5564\" width=\"12.769pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,40.956,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,48.964,0)\"></path></g></svg></span> be a simple graph with vertex set <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.35121 8.8423\" width=\"9.35121pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-87\"></use></g></svg> and edge set <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.13765 8.68572\" width=\"8.13765pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-70\"></use></g></svg>.</span> In a graph <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg>,</span> a subset of edges denoted by <svg height=\"9.09021pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.84467 14.0879 9.09021\" width=\"14.0879pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> is referred to as an edge-dominating set of <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg> if every edge that is not in <svg height=\"9.09021pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.84467 14.0879 9.09021\" width=\"14.0879pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-14\"></use></g></svg> is incident to at least one member of <span><svg height=\"9.09021pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.84467 14.0879 9.09021\" width=\"14.0879pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-14\"></use></g></svg>.</span> A set <span><svg height=\"9.96448pt\" style=\"vertical-align:-1.11981pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.84467 25.199 9.96448\" width=\"25.199pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-14\"></use></g><g transform=\"matrix(.013,0,0,-0.013,17.568,0)\"></path></g></svg><span></span><svg height=\"9.96448pt\" style=\"vertical-align:-1.11981pt\" version=\"1.1\" viewbox=\"28.7811838 -8.84467 8.218 9.96448\" width=\"8.218pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,28.831,0)\"><use xlink:href=\"#g113-70\"></use></g></svg></span> is the locating edge-dominating set if for every two edges <span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 13.227 12.4698\" width=\"13.227pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,5.317,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,10.263,0)\"></path></g></svg><span></span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"15.3571838 -9.28833 20.759 12.4698\" width=\"20.759pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,15.407,0)\"><use xlink:href=\"#g113-102\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,20.724,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,29.302,0)\"></path></g></svg><span></span><span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"39.7481838 -9.28833 44.703 12.4698\" width=\"44.703pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,39.798,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,44.296,0)\"><use xlink:href=\"#g113-70\"></use></g><g transform=\"matrix(.013,0,0,-0.013,55.21,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,65.746,0)\"><use xlink:href=\"#g198-14\"></use></g><g transform=\"matrix(.013,0,0,-0.013,79.682,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>,</span></span> the sets <span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 40.706 12.4698\" width=\"40.706pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,10.91,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,15.408,0)\"><use xlink:href=\"#g113-102\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,20.725,3.132)\"><use xlink:href=\"#g50-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.672,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,33.075,0)\"></path></g></svg><span></span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"43.5621838 -9.28833 14.207 12.4698\" width=\"14.207pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,43.612,0)\"><use xlink:href=\"#g198-14\"></use></g></svg></span> and <span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 40.706 12.4698\" width=\"40.706pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-79\"></use></g><g transform=\"matrix(.013,0,0,-0.013,10.91,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,15.408,0)\"><use xlink:href=\"#g113-102\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,20.725,3.132)\"><use xlink:href=\"#g50-51\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.672,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,33.075,0)\"><use xlink:href=\"#g117-60\"></use></g></svg><span></span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"43.5621838 -9.28833 14.207 12.4698\" width=\"14.207pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,43.612,0)\"><use xlink:href=\"#g198-14\"></use></g></svg></span> are nonempty and different. The edge domination number <svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 29.3663 12.7178\" width=\"29.3663pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,5.668,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,11.29,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,15.788,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,24.681,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> of <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg> is the minimum cardinality of all edge-dominating sets of <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg>.</span> The purpose of this study is to determine the locating edge domination number of certain types of claw-free cubic graphs.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/1182858","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a simple graph with vertex set and edge set . In a graph , a subset of edges denoted by is referred to as an edge-dominating set of if every edge that is not in is incident to at least one member of . A set is the locating edge-dominating set if for every two edges , the sets and are nonempty and different. The edge domination number of is the minimum cardinality of all edge-dominating sets of . The purpose of this study is to determine the locating edge domination number of certain types of claw-free cubic graphs.
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