Geodesic Geometry on Graphs

Daniel Cizma*, Nati Linial

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We investigate a graph theoretic analog of geodesic geometry. In a graph G= (V, E) we consider a system of paths P= { Pu,v: u, v∈ V} where Pu,v connects vertices u and v. This system is consistent in that if vertices y, z are in Pu,v, then the subpath of Pu,v between them coincides with Py,z. A map w: E→ (0 , ∞) is said to induceP if for every u, v∈ V the path Pu,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 languageAmerican English
Pages (from-to)298-347
Number of pages50
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - Jul 2022

Bibliographical note

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


  • Graph metrizability
  • Path systems
  • Shortest paths


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