{"title":"Strong Edge Geodetic Problem on Complete Multipartite Graphs and Some Extremal Graphs for the Problem","authors":"","doi":"10.1007/s41980-023-00849-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A set of vertices <em>X</em> of a graph <em>G</em> is a strong edge geodetic set if, to any pair of vertices from <em>X</em>, we can assign one (or zero) shortest path between them, such that every edge of <em>G</em> is contained in at least one on these paths. The cardinality of a smallest strong edge geodetic set of <em>G</em> is the strong edge geodetic number <span> <span>\\(\\mathrm{sg_e}(G)\\)</span> </span> of <em>G</em>. In this paper, the strong edge geodetic number of complete multipartite graphs is determined. Graphs <em>G</em> with <span> <span>\\(\\mathrm{sg_e}(G) = n(G)\\)</span> </span> are characterized and <span> <span>\\(\\mathrm{sg_e}\\)</span> </span> is determined for Cartesian products <span> <span>\\(P_n\\,\\square \\, K_m\\)</span> </span>. The latter result in particular corrects an error from the literature.</p>","PeriodicalId":9395,"journal":{"name":"Bulletin of The Iranian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of The Iranian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s41980-023-00849-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A set of vertices X of a graph G is a strong edge geodetic set if, to any pair of vertices from X, we can assign one (or zero) shortest path between them, such that every edge of G is contained in at least one on these paths. The cardinality of a smallest strong edge geodetic set of G is the strong edge geodetic number \(\mathrm{sg_e}(G)\) of G. In this paper, the strong edge geodetic number of complete multipartite graphs is determined. Graphs G with \(\mathrm{sg_e}(G) = n(G)\) are characterized and \(\mathrm{sg_e}\) is determined for Cartesian products \(P_n\,\square \, K_m\). The latter result in particular corrects an error from the literature.
摘要 如果我们能为 X 的任意一对顶点指定一条（或零条）最短路径，使得 G 的每条边都至少包含在其中一条路径中，那么图 G 的顶点集合 X 就是强边大地集。G 的最小强边大地集的心数就是 G 的强边大地数 \(\mathrm{sg_e}(G)\)。本文描述了具有 \(\mathrm{sg_e}(G) = n(G)\) 的图 G，并确定了笛卡尔积 \(P_n\,\square\, K_m\) 的 \(\mathrm{sg_e}\) 。后一个结果特别纠正了文献中的一个错误。
期刊介绍：
The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.
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