Sharmia H. Kaida, Kaimar Jay S. Maharajul, Javier A. Hassan, Ladznar S. Laja, Abdurajan B. Lintasan, Aljon A. Pablo
{"title":"Certified Hop Independence: Properties and Connections with other Variants of Independence","authors":"Sharmia H. Kaida, Kaimar Jay S. Maharajul, Javier A. Hassan, Ladznar S. Laja, Abdurajan B. Lintasan, Aljon A. Pablo","doi":"10.29020/nybg.ejpam.v17i1.5044","DOIUrl":null,"url":null,"abstract":"Let G be a graph. Then B ⊆ V (G) is called a certified hop independent set of G if for every a, b ∈ B, dG(a, b) ̸= 2 and for every x ∈ B has either zero or at least two neighbors in V (G) \\ B. The maximum cardinality among all certified hop independent sets in G, denoted by αch(G), is called the certified hop independence number of G. In this paper, we initiate the study of certified hop independence in graphs and we establish some of its properties. We give realization results involving hop independence and certified hop independence parameters, and we show that the difference between these two parameters can be made arbitrarily large. We characterize certified hop independent sets in some graphs and we use these results to obtain the exact values or bounds of the parameter. Moreover, we show that the certified hop independence and independence parameters are incomparable.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"196 6","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.29020/nybg.ejpam.v17i1.5044","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 G be a graph. Then B ⊆ V (G) is called a certified hop independent set of G if for every a, b ∈ B, dG(a, b) ̸= 2 and for every x ∈ B has either zero or at least two neighbors in V (G) \ B. The maximum cardinality among all certified hop independent sets in G, denoted by αch(G), is called the certified hop independence number of G. In this paper, we initiate the study of certified hop independence in graphs and we establish some of its properties. We give realization results involving hop independence and certified hop independence parameters, and we show that the difference between these two parameters can be made arbitrarily large. We characterize certified hop independent sets in some graphs and we use these results to obtain the exact values or bounds of the parameter. Moreover, we show that the certified hop independence and independence parameters are incomparable.
假设 G 是一个图。如果对于每个 a、b ∈ B,dG(a, b) ̸= 2,并且对于每个 x∈ B 在 V (G) \ B 中要么为零要么至少有两个邻居,那么 B ⊆ V (G) 称为 G 的认证跳独立集。G 中所有经认证的跳转独立集的最大心数,用 αch(G) 表示,称为 G 的经认证的跳转独立数。在本文中,我们开始研究图中经认证的跳转独立,并建立了它的一些属性。我们给出了涉及跳转独立性和认证跳转独立性参数的实现结果,并证明这两个参数之间的差值可以任意变大。我们描述了一些图中经认证的跳转独立集的特征,并利用这些结果获得了参数的精确值或边界。此外,我们还证明了经认证的跳转独立性和独立性参数是不可比的。
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.