On the hyperbolicity of Delaunay triangulations

If $ X $ is a geodesic metric space and $ x_1, x_2, x_3\in X $, a geodesic triangle $ T = \{x_1, x_2, x_3\} $ is the union of the three geodesics $ [x_1 x_2] $, $ [x_2 x_3] $ and $ [x_3 x_1] $ in $ X $. The space $ X $ is hyperbolic if there exists a constant $ \delta \ge 0 $ such that any side of a...

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Bibliographic Details
Published inAIMS mathematics Vol. 8; no. 12; pp. 28780 - 28790
Main Authors Carballosa, Walter, Rodríguez, José M., Sigarreta, José M.
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2023
Subjects
delaunay triangulation
hyperbolic graphs
tessellation graphs
voronoi graph
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ISSN2473-6988
2473-6988
DOI10.3934/math.20231474

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Summary:If $ X $ is a geodesic metric space and $ x_1, x_2, x_3\in X $, a geodesic triangle $ T = \{x_1, x_2, x_3\} $ is the union of the three geodesics $ [x_1 x_2] $, $ [x_2 x_3] $ and $ [x_3 x_1] $ in $ X $. The space $ X $ is hyperbolic if there exists a constant $ \delta \ge 0 $ such that any side of any geodesic triangle in $ X $ is contained in the $ \delta $-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.20231474