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Unit Ball Graphs on Geodesic Spaces

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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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Acknowledgements

The authors would like to thank the reviewers for valuable suggestions and comments, which were very helpful in making this paper more readable.

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Authors and Affiliations

  1. Faculty of Engineering, Nippon Bunri University, Oita, 870-0316, Japan

    Masamichi Kuroda

  2. Department of Education, Hokkaido University of Education, Hokkaido, 070-8621, Japan

    Shuhei Tsujie

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  1. Masamichi Kuroda
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  2. Shuhei Tsujie
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Correspondence to Shuhei Tsujie.

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Kuroda, M., Tsujie, S. Unit Ball Graphs on Geodesic Spaces. Graphs and Combinatorics 37, 111–125 (2021). https://doi.org/10.1007/s00373-020-02231-3

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  • DOI: https://doi.org/10.1007/s00373-020-02231-3

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