Shallow Minors, Graph Products, and Beyond-Planar Graphs

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-03-13 DOI:10.1137/22m1540296

Robert Hickingbotham, David R. Wood

{"title":"Shallow Minors, Graph Products, and Beyond-Planar Graphs","authors":"Robert Hickingbotham, David R. Wood","doi":"10.1137/22m1540296","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1057-1089, March 2024. <br/> Abstract. The planar graph product structure theorem of Dujmović et al. [J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, [math]-planar, [math]-cluster planar, fan-planar, and [math]-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial [math]-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that [math]-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1540296","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1057-1089, March 2024.
Abstract. The planar graph product structure theorem of Dujmović et al. [J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, [math]-planar, [math]-cluster planar, fan-planar, and [math]-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial [math]-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that [math]-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.

查看原文本刊更多论文

浅小数、图形积和超平面图形

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 1057-1089 页，2024 年 3 月。摘要。Dujmović 等人的平面图乘积结构定理[J. ACM, 67 (2020), 22]指出，每个平面图都是有界树宽的图与路径的强乘积的子图。这一结果是解决队列布局、非重复着色、居中着色和邻接标记方案等重要开放问题的关键工具。在本文中，我们扩展了这一研究方向，利用浅小数证明了几类超平面图的类似乘积结构定理。推动我们工作的关键观点是，许多超平面图可以描述为平面图与小完整图的强积的浅次要图。特别是，我们证明了有界度平面图、[math]-平面图、[math]-簇平面图、扇形平面图和[math]-扇形束平面图的幂具有这样的浅次要结构。利用新旧结果的结合，我们推导出这些类具有有界队列数、有界非重复色度数、多项式[math]中心色度数、线性强着色数和立方弱着色数。此外，我们还证明了[math]空隙平面图至少具有指数局部树宽，因此不能被描述为具有有界树宽的图与路径的强积的子图。

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

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.