平面跨路径重构的进一步连接结果

arXiv - CS - Discrete Mathematics Pub Date : 2024-06-28 DOI:arxiv-2407.00244

Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier

{"title":"平面跨路径重构的进一步连接结果","authors":"Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier","doi":"arxiv-2407.00244","DOIUrl":null,"url":null,"abstract":"Given a finite set $ S $ of points, we consider the following reconfiguration\ngraph. The vertices are the plane spanning paths of $ S $ and there is an edge\nbetween two vertices if the two corresponding paths differ by two edges (one\nremoved, one added). Since 2007, this graph is conjectured to be connected but\nno proof has been found. In this paper, we prove several results to support the\nconjecture. Mainly, we show that if all but one point of $ S $ are in convex\nposition, then the graph is connected with diameter at most $ 2 | S | $ and\nthat for $ | S | \\geq 3 $ every connected component has at least $ 3 $\nvertices.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"729 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Further Connectivity Results on Plane Spanning Path Reconfiguration\",\"authors\":\"Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier\",\"doi\":\"arxiv-2407.00244\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a finite set $ S $ of points, we consider the following reconfiguration\\ngraph. The vertices are the plane spanning paths of $ S $ and there is an edge\\nbetween two vertices if the two corresponding paths differ by two edges (one\\nremoved, one added). Since 2007, this graph is conjectured to be connected but\\nno proof has been found. In this paper, we prove several results to support the\\nconjecture. Mainly, we show that if all but one point of $ S $ are in convex\\nposition, then the graph is connected with diameter at most $ 2 | S | $ and\\nthat for $ | S | \\\\geq 3 $ every connected component has at least $ 3 $\\nvertices.\",\"PeriodicalId\":501216,\"journal\":{\"name\":\"arXiv - CS - Discrete Mathematics\",\"volume\":\"729 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.00244\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.00244","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

给定一个有限的点集合 $ S $，我们考虑下面的重组图。顶点是 $ S $ 的平面跨越路径，如果两条对应路径相差两条边（一条删除，一条添加），则两个顶点之间有一条边。自 2007 年以来，人们一直猜测这个图是连通的，但没有找到证明。在本文中，我们证明了支持该猜想的几个结果。主要是，我们证明了如果除了一个点之外，$ S $ 的所有点都在凸点上，那么这个图是连通的，直径最多为 $ 2 | S | $，并且对于 $ | S |\geq 3 $，每个连通的部分至少有 $ 3 $ 个顶点。

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

微信好友朋友圈 QQ好友复制链接

本刊更多论文

Further Connectivity Results on Plane Spanning Path Reconfiguration

Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one removed, one added). Since 2007, this graph is conjectured to be connected but no proof has been found. In this paper, we prove several results to support the conjecture. Mainly, we show that if all but one point of $ S $ are in convex position, then the graph is connected with diameter at most $ 2 | S | $ and that for $ | S | \geq 3 $ every connected component has at least $ 3 $ vertices.

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

来源期刊

arXiv - CS - Discrete Mathematics

自引率

0.00%

发文量

期刊最新文献

Reconfiguration of labeled matchings in triangular grid graphs Decision problems on geometric tilings Ants on the highway A sequential solution to the density classification task using an intermediate alphabet Complexity of Deciding the Equality of Matching Numbers