Connected coalitions in graphs

IF 0.5 4区 数学 Q3 MATHEMATICS
S. Alikhani, D. Bakhshesh, H. Golmohammadi, E. Konstantinova
{"title":"Connected coalitions in graphs","authors":"S. Alikhani, D. Bakhshesh, H. Golmohammadi, E. Konstantinova","doi":"10.7151/dmgt.2509","DOIUrl":null,"url":null,"abstract":"The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $\\psi$ = $\\{V_1, V_2,..., V_k \\}$ such that every set $V_i \\in \\psi$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\\in \\psi$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $\\delta(G)=1$ and with no full vertex, it holds that $CC(G)<n$. Furthermore, we show that for any tree $T$, $CC(T)=2$. Finally, we present two polynomial-time algorithms that for a given connected graph $G$ of order $n$ determine whether $CC(G)=n$ or $CC(G)=n-1$.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discussiones Mathematicae Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2509","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $\psi$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in \psi$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\in \psi$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $\delta(G)=1$ and with no full vertex, it holds that $CC(G)
图中的连通联盟
图$G=(V,E)$中的连通联盟由两个不相交的顶点集$V_{1}$和$V_{2}$组成,这两个顶点集都不是连通支配集,但其并集$V_{1}\cup V_{2}$是连通支配集。$n=|V|$阶图$G$中的连通联盟分区是顶点分区$\psi$=$\{V_1,V_2,…,V_k\}$,使得每个集合$V_i\in\psi$要么是由一个阶为$n-1$的单顶点组成的连通支配集,要么不是连通支配集而是与另一个不是连通支配集合的集合$V_j\in\psi形成连通联盟。连接联盟数,用$CC(G)$表示,是$G$的连接联盟分区的最大基数。本文首先对图中的连通联盟进行了研究,并给出了一些基本结果。准确地说,我们刻画了所有具有连通联盟划分的图。此外,我们证明了对于任何$n$阶的图$G$,$\delta(G)=1$并且没有全顶点,它认为$CC(G)
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
2.20
自引率
0.00%
发文量
22
审稿时长
53 weeks
期刊介绍: The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信