Geodesic Geometry on Graphs

Daniel Cizma; Nati Linial

doi:10.1007/s00454-021-00345-w

Geodesic Geometry on Graphs

Daniel Cizma^*, Nati Linial

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We investigate a graph theoretic analog of geodesic geometry. In a graph G= (V, E) we consider a system of paths P= { P_u_,_v: u, v∈ V} where P_u_,_v connects vertices u and v. This system is consistent in that if vertices y, z are in P_u_,_v, then the subpath of P_u_,_v between them coincides with P_y_,_z. A map w: E→ (0 , ∞) is said to induceP if for every u, v∈ V the path P_u_,_v is w-geodesic. We say that G is metrizable if every consistent path system is induced by some such w. As we show, metrizable graphs are very rare, whereas there exist infinitely many 2-connected metrizable graphs.

Original language	English
Pages (from-to)	298-347
Number of pages	50
Journal	Discrete and Computational Geometry
Volume	68
Issue number	1
DOIs	https://doi.org/10.1007/s00454-021-00345-w
State	Published - Jul 2022

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Graph metrizability
Path systems
Shortest paths

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Geodesic Geometry on Graphs

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Keywords

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