图中的总2-彩虹支配:复杂性和算法

IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Manjay Kumar, P. Venkata Subba Reddy
{"title":"图中的总2-彩虹支配:复杂性和算法","authors":"Manjay Kumar, P. Venkata Subba Reddy","doi":"10.1142/s0129054123500260","DOIUrl":null,"url":null,"abstract":"For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Total 2-Rainbow Domination in Graphs: Complexity and Algorithms\",\"authors\":\"Manjay Kumar, P. Venkata Subba Reddy\",\"doi\":\"10.1142/s0129054123500260\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.\",\"PeriodicalId\":50323,\"journal\":{\"name\":\"International Journal of Foundations of Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-10-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Foundations of Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1142/s0129054123500260\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Foundations of Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0129054123500260","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0

摘要

对于没有孤立顶点的简单无向图[公式:见文],满足以下两个条件的函数[公式:见文]称为[公式:见文]的总2-彩虹支配函数(T2RDF)。(i)对于所有[公式:见文],如果[公式:见文],则[公式:见文];(ii)每个[公式:见文]与[公式:见文]的顶点[公式:见文]相邻。[Formula: see text]的T2RDF [Formula: see text]的权重是值[Formula: see text]。总2彩虹控制数是一个T2RDF在[公式:见文本]上的最小权重,用[公式:见文本]表示。最小总2彩虹支配问题(MT2RDP)是在输入图中找到一个最小权重的T2RDF。在本文中,我们证明了判定星形凸二部图、梳状凸二部图、分裂图和平面图的[公式:见文]是否有最大权值的T2RDF的问题是np完全的。在积极的方面,我们证明了MT2RDP对于阈值图、链图和有界树宽度图是线性时间可解的。从近似的角度来看,我们表明MT2RDP不能在[公式:见文]内近似任何[公式:见文],除非[公式:见文],并提出[公式:见文]-近似算法。进一步,我们证明了MT2RDP对于最大度为4的图是apx完全的。其次,证明了控制问题和总2彩虹控制问题在计算复杂度方面是不等价的。最后,给出了MT2RDP的整数线性规划公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Total 2-Rainbow Domination in Graphs: Complexity and Algorithms
For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.
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
International Journal of Foundations of Computer Science
International Journal of Foundations of Computer Science 工程技术-计算机:理论方法
CiteScore
1.60
自引率
12.50%
发文量
63
审稿时长
3 months
期刊介绍: The International Journal of Foundations of Computer Science is a bimonthly journal that publishes articles which contribute new theoretical results in all areas of the foundations of computer science. The theoretical and mathematical aspects covered include: - Algebraic theory of computing and formal systems - Algorithm and system implementation issues - Approximation, probabilistic, and randomized algorithms - Automata and formal languages - Automated deduction - Combinatorics and graph theory - Complexity theory - Computational biology and bioinformatics - Cryptography - Database theory - Data structures - Design and analysis of algorithms - DNA computing - Foundations of computer security - Foundations of high-performance computing
×
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
0
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