The Structure of Metrizable Graphs
Combinatorics
2023-11-17 v1
Abstract
A consistent path system in a graph is an intersection-closed collection of paths, with exactly one path between any two vertices in . We call metrizable 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.
Cite
@article{arxiv.2311.09364,
title = {The Structure of Metrizable Graphs},
author = {Maria Chudnovsky and Daniel Cizma and Nati Linial},
journal= {arXiv preprint arXiv:2311.09364},
year = {2023}
}