A closer look at Hamiltonicity and domination through the lens of diameter and convexity

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS
R. Mahendra Kumar, N. Sadagopan
{"title":"A closer look at Hamiltonicity and domination through the lens of diameter and convexity","authors":"R. Mahendra Kumar,&nbsp;N. Sadagopan","doi":"10.1007/s00236-024-00461-z","DOIUrl":null,"url":null,"abstract":"<div><p>A bipartite graph <i>G</i>(<i>X</i>, <i>Y</i>) is called a star-convex bipartite graph with convexity on <i>X</i> if there is an associated star <i>T</i>(<i>X</i>, <i>F</i>), such that for each vertex in <i>Y</i>, its neighborhood in <i>X</i> induces a subtree in <i>T</i>. A graph <i>G</i> is said to be a split graph if <i>G</i> can be partitioned into a clique (<i>K</i>) and an independent set (<i>I</i>). The objective of this study is twofold: (i) to strengthen the complexity results presented in Chen et al. (J Comb Optim 32(1):95–110, 2016) for the Hamiltonian cycle (HCYCLE), the Hamiltonian path (HPATH), and the Domination (DS) problems on star-convex bipartite graphs (ii) to reinforce the results of Müller (Discret Math 156(1–3):291–298, 1996) for HCYCLE, and HPATH on split graphs by introducing a convex ordering on one of the partitions (<i>K</i> or <i>I</i>). As part of our fine-grained analysis study with the diameter being the parameter, we first show that the diameter of star-convex bipartite graphs is at most six. Next, we observe that the reduction instances of Chen et al. (J Comb Optim 32(1):95–110, 2016) are star-convex bipartite graphs with at most diameter 4, and hence HCYCLE and HPATH are NP-complete on star-convex bipartite graphs with at most diameter 4. We strengthen this result and establish the following results on star-convex bipartite graphs: (i) HCYCLE is NP-complete for diameter 3, and polynomial-time solvable for diameters 2, 5, and 6 (a transformation in complexity: P to NPC to P) (ii) HPATH is polynomial-time solvable for diameter 2, and NP-Complete, otherwise (a dichotomy). Further, with convexity being the parameter, for split graphs with convexity on <i>K</i> (resp. <i>I</i>), we show that HCYCLE and HPATH are NP-complete on star-convex (resp. comb) split graphs with convexity on <i>K</i> (resp. <i>I</i>). Further, we show that HCYCLE is NP-complete on <span>\\(k_{1,r}\\)</span>-free star-convex split graphs with convexity on <i>I</i>, <span>\\(r\\ge 6\\)</span>. On the positive side, we show that for <span>\\(K_{1,5}\\)</span>-free star-convex split graphs with convexity on <i>I</i>, HCYCLE is polynomial-time solvable. Thus, we establish a dichotomy for HCYCLE on star-convex split graphs with convexity on <i>I</i>. We further show that the dominating set problem (DS) and its variants (resp. Connected, Total, Outer-Connected, and Dominating biclique) are NP-complete on star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6). On the parameterized complexity front, we prove that the parameterized version of the domination problem and its variants, with the parameter being the solution size, is not fixed-parameter tractable for star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6), whereas it is fixed-parameter tractable when the parameter is the number of leaves in the associated star. Further, we show that for star-convex bipartite graphs with diameters 5, and 6, the domination problem and its variants cannot be approximated within <span>\\((1-\\epsilon )\\)</span> ln <i>n</i> unless NP <span>\\(\\subseteq TIME(2^{n^ {o(1)}})\\)</span>.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Informatica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00236-024-00461-z","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0

Abstract

A bipartite graph G(XY) is called a star-convex bipartite graph with convexity on X if there is an associated star T(XF), such that for each vertex in Y, its neighborhood in X induces a subtree in T. A graph G is said to be a split graph if G can be partitioned into a clique (K) and an independent set (I). The objective of this study is twofold: (i) to strengthen the complexity results presented in Chen et al. (J Comb Optim 32(1):95–110, 2016) for the Hamiltonian cycle (HCYCLE), the Hamiltonian path (HPATH), and the Domination (DS) problems on star-convex bipartite graphs (ii) to reinforce the results of Müller (Discret Math 156(1–3):291–298, 1996) for HCYCLE, and HPATH on split graphs by introducing a convex ordering on one of the partitions (K or I). As part of our fine-grained analysis study with the diameter being the parameter, we first show that the diameter of star-convex bipartite graphs is at most six. Next, we observe that the reduction instances of Chen et al. (J Comb Optim 32(1):95–110, 2016) are star-convex bipartite graphs with at most diameter 4, and hence HCYCLE and HPATH are NP-complete on star-convex bipartite graphs with at most diameter 4. We strengthen this result and establish the following results on star-convex bipartite graphs: (i) HCYCLE is NP-complete for diameter 3, and polynomial-time solvable for diameters 2, 5, and 6 (a transformation in complexity: P to NPC to P) (ii) HPATH is polynomial-time solvable for diameter 2, and NP-Complete, otherwise (a dichotomy). Further, with convexity being the parameter, for split graphs with convexity on K (resp. I), we show that HCYCLE and HPATH are NP-complete on star-convex (resp. comb) split graphs with convexity on K (resp. I). Further, we show that HCYCLE is NP-complete on \(k_{1,r}\)-free star-convex split graphs with convexity on I, \(r\ge 6\). On the positive side, we show that for \(K_{1,5}\)-free star-convex split graphs with convexity on I, HCYCLE is polynomial-time solvable. Thus, we establish a dichotomy for HCYCLE on star-convex split graphs with convexity on I. We further show that the dominating set problem (DS) and its variants (resp. Connected, Total, Outer-Connected, and Dominating biclique) are NP-complete on star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6). On the parameterized complexity front, we prove that the parameterized version of the domination problem and its variants, with the parameter being the solution size, is not fixed-parameter tractable for star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6), whereas it is fixed-parameter tractable when the parameter is the number of leaves in the associated star. Further, we show that for star-convex bipartite graphs with diameters 5, and 6, the domination problem and its variants cannot be approximated within \((1-\epsilon )\) ln n unless NP \(\subseteq TIME(2^{n^ {o(1)}})\).

Abstract Image

Abstract Image

从直径和凸度的角度细看汉密尔顿性和支配性
如果存在一个相关星 T(X,F),使得 Y 中的每个顶点在 X 中的邻域都在 T 中诱导出一个子树,那么一个双态图 G(X, Y)被称为在 X 上具有凸性的星凸双态图。本研究的目的有两个:(i) 强化 Chen 等人的复杂性结果(《J Comb Optim》32(1)(J Comb Optim 32(1):95-110, 2016) 中提出的星凸双方形图上的汉密尔顿循环(HCYCLE)、汉密尔顿路径(HPATH)和主宰(DS)问题的复杂度结果(ii)通过在其中一个分区(K 或 I)上引入凸排序,加强 Müller (Discret Math 156(1-3):291-298, 1996) 中提出的分裂图上的汉密尔顿循环和汉密尔顿路径问题的复杂度结果。作为以直径为参数的细粒度分析研究的一部分,我们首先证明星凸双artite图的直径最多为6。接下来,我们观察到 Chen 等人(J Comb Optim 32(1):95-110, 2016)的还原实例是直径最多为 4 的星凸双叉图,因此 HCYCLE 和 HPATH 在直径最多为 4 的星凸双叉图上是 NP-完备的。我们强化了这一结果,并在星凸双叉图上建立了以下结果:(i) HCYCLE 对于直径为 3 的图是 NP-完全的,而对于直径为 2、5 和 6 的图是多项式时间可解的(复杂性的转换:P 到 NPC 到 P) (ii) HPATH 对于直径为 2 的图是多项式时间可解的,否则是 NP-完全的(二分法)。此外,以凸性为参数,对于 K(或 I)上具有凸性的分裂图,我们证明 HCYCLE 和 HPATH 在 K(或 I)上具有凸性的星凸(或梳状)分裂图上是 NP-完全的。此外,我们还证明了 HCYCLE 在 I 上具有凸性的(k_{1,r}\)无星凸分裂图上是 NP-完全的。从正面来看,我们证明了对于 I 上具有凸性的(K_{1,5}\)无星凸分裂图,HCYCLE 是多项式时间可解的。我们进一步证明,在直径为 3(直径为 5 和直径为 6)的星凸双瓣图上,支配集问题(Dominating Set problem,DS)及其变体(分别为连通问题、全连通问题、外连通问题和支配双瓣问题)是 NP-完全的。在参数化复杂性方面,我们证明了参数化版本的支配问题及其变体(参数为解大小)对于直径为 3(分别为直径为 5 和直径为 6)的星凸双瓣图来说不是固定参数可处理的,而当参数为相关星中的叶片数时则是固定参数可处理的。此外,我们还证明了对于直径为 5 和 6 的星凸双啮合图,支配问题及其变体无法在 \((1-\epsilon )\) ln n 内近似,除非 NP \(\subseteq TIME(2^{n^ {o(1)}})\).
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
Acta Informatica
Acta Informatica 工程技术-计算机:信息系统
CiteScore
2.40
自引率
16.70%
发文量
24
审稿时长
>12 weeks
期刊介绍: Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics. Topics of interest include: • semantics of programming languages • models and modeling languages for concurrent, distributed, reactive and mobile systems • models and modeling languages for timed, hybrid and probabilistic systems • specification, program analysis and verification • model checking and theorem proving • modal, temporal, first- and higher-order logics, and their variants • constraint logic, SAT/SMT-solving techniques • theoretical aspects of databases, semi-structured data and finite model theory • theoretical aspects of artificial intelligence, knowledge representation, description logic • automata theory, formal languages, term and graph rewriting • game-based models, synthesis • type theory, typed calculi • algebraic, coalgebraic and categorical methods • formal aspects of performance, dependability and reliability analysis • foundations of information and network security • parallel, distributed and randomized algorithms • design and analysis of algorithms • foundations of network and communication protocols.
×
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