  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.

Combinatorics