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":15840,"journal":{"name":"Journal of Function Spaces","volume":"3 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Function Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/1182858","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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.
期刊介绍：
Journal of Function Spaces (formerly titled Journal of Function Spaces and Applications) publishes papers on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. As well as original research, Journal of Function Spaces also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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