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.

考虑测地空间中的有限多个点。如果两点的距离小于固定阈值，则我们将这两点视为“近”。将附近的点与边连接起来，我们在点上获得一个简单的图，称为单位球图。如果空间是实线，则称为单位间隔图。测地空间上的单位球图以图形的形式描述了空间的几何特征。在本文中，我们证明，当且仅当空间为\（{\\ mathbb {R}} \）时，测地空间上的每个单位球图都是（强烈）弦的-tree且测地空间上的每个单位球图都是自由的（爪形，净值），且仅当该空间是最大尺寸为1的连通流形时。作为推论，我们证明了单位球图的本质是表征实线和单位圆。