定位总支配集三分之二猜想的进展

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-07-24 DOI:10.1016/j.disc.2024.114176

{"title":"定位总支配集三分之二猜想的进展","authors":"","doi":"10.1016/j.disc.2024.114176","DOIUrl":null,"url":null,"abstract":"<div>We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set S of vertices of a graph G is a locating-total dominating set if every vertex of G has a neighbor in S, and if any two vertices outside S have distinct neighborhoods within S. The smallest size of such a set is denoted by <math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math>. It has been conjectured that <math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math> holds for every twin-free graph G of order n without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.</div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Progress towards the two-thirds conjecture on locating-total dominating sets\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114176\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set S of vertices of a graph G is a locating-total dominating set if every vertex of G has a neighbor in S, and if any two vertices outside S have distinct neighborhoods within S. The smallest size of such a set is denoted by <math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math>. It has been conjectured that <math><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math> holds for every twin-free graph G of order n without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.</div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24003078\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003078","RegionNum":3,"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

摘要

我们研究图中最优定位总支配集大小的上限。如果图 G 的每个顶点在 S 中都有一个邻居，并且 S 外的任何两个顶点在 S 中都有不同的邻域，那么图 G 的顶点集合 S 就是定位-总支配集。有人猜想，γtL(G)≤2n3 对于每一个无孤立顶点的 n 阶无孪生图 G 都成立。我们证明该猜想对于共方图、分裂图、块图和次立方图都成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Progress towards the two-thirds conjecture on locating-total dominating sets

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set S of vertices of a graph G is a locating-total dominating set if every vertex of G has a neighbor in S, and if any two vertices outside S have distinct neighborhoods within S. The smallest size of such a set is denoted by $γ_{t}^{L} (G)$ . It has been conjectured that $γ_{t}^{L} (G) \leq \frac{2 n}{3}$ holds for every twin-free graph G of order n without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.