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Geodesic Geometry on Graphs

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Abstract

We investigate a graph theoretic analog of geodesic geometry. In a graph \(G=(V,E)\) we consider a system of paths \(\mathcal {P}=\{P_{u,v}:u,v\in V\}\) 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 \(w:E\rightarrow (0,\infty )\) is said to induce \(\mathcal {P}\) if for every \(u, v\in V\) 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.

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Notes

  1. In his terminology strongly metrizable.

  2. https://github.com/dcizma1/testing-graph-metrizability.

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Authors and Affiliations

  1. Department of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Daniel Cizma

  2. School of Computer Science and Engineering, Hebrew University, Jerusalem, 91904, Israel

    Nati Linial

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  1. Daniel Cizma
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  2. Nati Linial
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Correspondence to Daniel Cizma.

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Appendix A: Certificates of Non-Metrizability

Appendix A: Certificates of Non-Metrizability

For each graph G in Fig. 22 we give a path system in G along with a system of inequalities a weight function inducing this path system must satisfy. In each case, these inequalities imply at least one edge in the graph must have a non-positive weight, showing the graph in not metrizable.

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Cizma, D., Linial, N. Geodesic Geometry on Graphs. Discrete Comput Geom 68, 298–347 (2022). https://doi.org/10.1007/s00454-021-00345-w

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