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arXiv · 1809.08608

Unit Ball Graphs on Geodesic Spaces

Masamichi Kuroda·Shuhei Tsujie
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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.

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Masamichi Kuroda, Shuhei Tsujie. 2018-09-23. Unit Ball Graphs on Geodesic Spaces. https://doi.org/10.1007/s00373-020-02231-3

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