Unit Ball Graphs on Geodesic Spaces

TL;DR

This paper characterizes the geometric properties of geodesic spaces through the structure of their unit ball graphs, revealing that certain graph properties correspond precisely to the space being an $\mathbb{R}$-tree or a 1-dimensional manifold.

Contribution

It establishes a correspondence between graph-theoretic properties of unit ball graphs and the geometric structure of the underlying space, specifically characterizing $\mathbb{R}$-trees and 1-dimensional manifolds.

Findings

Unit ball graphs on geodesic spaces are (strongly) chordal iff the space is an $\mathbb{R}$-tree.

Unit ball graphs are (claw, net)-free iff the space is a connected 1-dimensional manifold.

The collection of unit ball graphs characterizes the real line and the unit circle.

Abstract

Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an $ \mathbb{R} $-tree and that every unit ball graph on a geodesic space is (claw, net)-free if and only if the space is a connected manifold of dimension at most $ 1 $. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.