Meiqin Wei , Bohua Fan , Changhong Lu , Jun Yue , Jinfeng Liu
{"title":"The edge metric dimensions of convex polytopes","authors":"Meiqin Wei , Bohua Fan , Changhong Lu , Jun Yue , Jinfeng Liu","doi":"10.1016/j.dam.2025.07.027","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> be a connected graph. A vertex <span><math><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow></math></span> distinguishes the edge pair <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>E</mi></mrow></math></span> if the distances from <span><math><mi>x</mi></math></span> to <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are distinct. A vertex subset <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is an edge metric generator of <span><math><mi>G</mi></math></span> if any pair of edges in <span><math><mi>E</mi></math></span> can be distinguished by some element of <span><math><mi>S</mi></math></span>. The minimum size of an edge metric generator of <span><math><mi>G</mi></math></span> is called the edge metric dimension of <span><math><mi>G</mi></math></span> and denoted by <span><math><mrow><mi>e</mi><mi>d</mi><mi>i</mi><mi>m</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we determine the exact values of the edge metric dimensions for some convex polytopes and generalized convex polytopes, which further emphasize the fact that there are families of convex polytopes having greater edge metric dimensions than their metric dimensions. The proof methods in this paper are constructive and they can be implemented through algorithms.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 294-306"},"PeriodicalIF":1.0000,"publicationDate":"2025-07-30","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/S0166218X25004184","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
Let be a connected graph. A vertex distinguishes the edge pair if the distances from to and are distinct. A vertex subset is an edge metric generator of if any pair of edges in can be distinguished by some element of . The minimum size of an edge metric generator of is called the edge metric dimension of and denoted by . In this paper, we determine the exact values of the edge metric dimensions for some convex polytopes and generalized convex polytopes, which further emphasize the fact that there are families of convex polytopes having greater edge metric dimensions than their metric dimensions. The proof methods in this paper are constructive and they can be implemented through algorithms.
期刊介绍:
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
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