# The Size of the Open Sphere of Influence Graph in *L*_{∞} Metric Spaces

Abstract

Conference paper

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## Abstract

Let *V* be a set of distinct points in some metric space. For each point *x* ∈ *V*, let *r* _{ x } be the distance from *x* to its nearest neighbour, and let *s* _{ x } be the open ball centered at *x* with radius equal to the distance from *x* to its nearest neighbour. We refer to these balls as the *spheres of influence* of the set *V.* The *open sphere of influence graph* on *V* is defined as the graph where (*x*,*y*) is an edge if and only if *s* _{ x } and *s* _{ y } intersect.

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## References

- [1]David Avis and Joe Horton. Remarks on the sphere-of-influence graph.
*Discrete Geometry and Convexity*, 440:323–327, 1985.MathSciNetGoogle Scholar - [2]Leonidas Guibas, János Pach, and Micha Sharir. Sphere-of-influence graphs in higher dimensions.
*Colloquia Mathematica Societatis János Bolyai*, 63:131–137, 1994.MathSciNetGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1998