Unit Ball Graphs on Geodesic Spaces

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.