论[数学]最短路径可覆盖的图

Pub Date : 2024-06-10 DOI:10.1137/23m1564511

Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca

{"title":"论[数学]最短路径可覆盖的图","authors":"Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca","doi":"10.1137/23m1564511","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024. <br/> Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Graphs Coverable by [math] Shortest Paths\",\"authors\":\"Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca\",\"doi\":\"10.1137/23m1564511\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024. <br/> Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/23m1564511\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1564511","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

SIAM 离散数学杂志》第 38 卷第 2 期第 1840-1862 页，2024 年 6 月。摘要。我们证明，如果一个无向图[math]的边或顶点可以被[math]最短路径覆盖，那么[math]的路径宽度是由[math]的单指数函数上界的。作为推论，我们证明带终端的等距路径覆盖问题（给定一个图[math]和一组称为终端的[math]对顶点，问[math]是否能被[math]最短路径覆盖，每条路径连接一对终端）是关于终端数的 FPT 问题。类似的问题 "有终端的强大地集"（给定一个图[math]和一组[math]终端，问是否存在覆盖[math]的[math]最短路径，每条路径连接一对不同的终端）也是如此。此外，这意味着相关问题等距路径覆盖和强大地集（定义类似，但终点集不是输入的一部分）在参数[math]方面是XP的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

On Graphs Coverable by [math] Shortest Paths

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024.
Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].

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