An improved spectral lower bound of treewidth

IF 0.7 4区计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS

Information Processing Letters Pub Date : 2024-10-17 DOI:10.1016/j.ipl.2024.106536

Tatsuya Gima , Tesshu Hanaka , Kohei Noro , Hirotaka Ono , Yota Otachi

{"title":"An improved spectral lower bound of treewidth","authors":"Tatsuya Gima , Tesshu Hanaka , Kohei Noro , Hirotaka Ono , Yota Otachi","doi":"10.1016/j.ipl.2024.106536","DOIUrl":null,"url":null,"abstract":"<div><div>We show that for every <em>n</em>-vertex graph with at least one edge, its treewidth is greater than or equal to <span><math><mi>n</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><mo>(</mo><mi>Δ</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where Δ and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [<em>Inf. Process. Lett.</em>, 2003] and the subsequent one by the authors of the present paper [<em>IEICE Trans. Inf. Syst.</em>, 2024]. The new lower bound is <em>almost</em> tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Information Processing Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020019024000668","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

We show that for every n-vertex graph with at least one edge, its treewidth is greater than or equal to

n λ_{2} / (Δ + λ_{2}) - 1

, where Δ and

λ_{2}

are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.

查看原文本刊更多论文

改进的树宽光谱下界

我们证明，对于每个至少有一条边的 n 顶点图，其树宽都大于或等于 nλ2/(Δ+λ2)-1，其中 Δ 和 λ2 分别是图的最大度和第二小拉普拉奇特征值。这个下界改进了 Chandran 和 Subramanian [Inf. Process. Lett.新的下界几乎是严密的，因为存在一个无限图族，使得下界只比族中每个图的树宽小 1。此外，利用类似的技术，我们还提出了最大和第二小拉普拉契亚特征值的树宽下界。

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

Information Processing Letters 工程技术-计算机：信息系统

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

7.3 months

期刊介绍： Information Processing Letters invites submission of original research articles that focus on fundamental aspects of information processing and computing. This naturally includes work in the broadly understood field of theoretical computer science; although papers in all areas of scientific inquiry will be given consideration, provided that they describe research contributions credibly motivated by applications to computing and involve rigorous methodology. High quality experimental papers that address topics of sufficiently broad interest may also be considered. Since its inception in 1971, Information Processing Letters has served as a forum for timely dissemination of short, concise and focused research contributions. Continuing with this tradition, and to expedite the reviewing process, manuscripts are generally limited in length to nine pages when they appear in print.