On the homotopy type of multipath complexes

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-12-09 DOI:10.1112/mtk.12235

Luigi Caputi, Carlo Collari, Sabino Di Trani, Jason P. Smith

{"title":"On the homotopy type of multipath complexes","authors":"Luigi Caputi, Carlo Collari, Sabino Di Trani, Jason P. Smith","doi":"10.1112/mtk.12235","DOIUrl":null,"url":null,"abstract":"<p>A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph <math>\n <semantics>\n <mi>G</mi>\n <annotation>${\\tt G}$</annotation>\n </semantics></math> is the simplicial complex whose faces are the multipaths of <math>\n <semantics>\n <mi>G</mi>\n <annotation>${\\tt G}$</annotation>\n </semantics></math>. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if <math>\n <semantics>\n <mi>G</mi>\n <annotation>${\\tt G}$</annotation>\n </semantics></math> is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of <math>\n <semantics>\n <mi>G</mi>\n <annotation>${\\tt G}$</annotation>\n </semantics></math> is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12235","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

A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph $G$ is the simplicial complex whose faces are the multipaths of $G$ . We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if $G$ is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of $G$ is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.

查看原文本刊更多论文

论多径复合体的同调类型

有向图中的多路径是路径的不相交联合。有向图 G ${\tt G}$ 的多径复合体是其面为 G ${\tt G}$ 的多径的简单复合体。我们计算了某些族有向图的多径复数的欧拉特征和相关的生成函数，其中包括传递锦标赛和完全二方图。我们证明，如果 G ${\tt G}$ 是线性图、多边形、小网格或反式锦标赛，那么 G ${\tt G}$ 的多径复合体的同调类型总是可收缩的或楔形的。我们引入了一种将有向图分解为动态区域的新技术，从而简化了同调计算。

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

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.