Geodesic Geometry on Graphs

TL;DR

This paper explores a graph-theoretic analog of geodesic geometry, characterizing when a graph's consistent path systems can be induced by edge weights, revealing that metrizable graphs are rare but infinitely many 2-connected examples exist.

Contribution

It introduces the concept of geodesic systems on graphs, defines metrizable graphs, and characterizes their rarity and abundance in 2-connected cases.

Findings

Metrizable graphs are very rare.

Infinitely many 2-connected metrizable graphs exist.

A new framework for geodesic geometry on graphs is proposed.

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 sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w: E\to(0,\infty)$ 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.