{"title":"The Structure of Metrizable Graphs","authors":"Maria Chudnovsky, Daniel Cizma, Nati Linial","doi":"10.1007/s00454-024-00685-3","DOIUrl":null,"url":null,"abstract":"<p>A <i>consistent path system</i> in a graph <i>G</i> is an intersection-closed collection of paths, with exactly one path between any two vertices in <i>G</i>. We call <i>G</i> <i>metrizable</i> if every consistent path system in it is the system of geodesic paths defined by assigning some positive lengths to its edges. We show that metrizable graphs are, in essence, subdivisions of a small family of basic graphs with additional compliant edges. In particular, we show that every metrizable graph with 11 vertices or more is outerplanar plus one vertex.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"24-25 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00685-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
A consistent path system in a graph G is an intersection-closed collection of paths, with exactly one path between any two vertices in G. We call Gmetrizable if every consistent path system in it is the system of geodesic paths defined by assigning some positive lengths to its edges. We show that metrizable graphs are, in essence, subdivisions of a small family of basic graphs with additional compliant edges. In particular, we show that every metrizable graph with 11 vertices or more is outerplanar plus one vertex.
图 G 中的一致路径系统是路径的交集-封闭集合，G 中任意两个顶点之间都有一条路径。如果图 G 中的每个一致路径系统都是通过为其边分配一些正长度而定义的大地路径系统，我们就称其为可元胞图。我们证明，可元胞图实质上是基本图的一个小族的细分，带有额外的符合边。特别是，我们证明了每一个有 11 个或更多顶点的可元胞图都是外平面加一个顶点。
期刊介绍：
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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