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Sphere-of-Influence Graphs in Normed Spaces

  • Márton Naszódi
  • János Pach
  • Konrad Swanepoel
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)

Abstract

We show that any k-th closed sphere-of-influence graph in a d-dimensional normed space has a vertex of degree less than \(5^d k\), thus obtaining a common generalization of results of Füredi and Loeb (Proc Am Math Soc 121(4):1063–1073, 1994 [1]) and Guibas et al. (Sphere-of-influence graphs in higher dimensions, Intuitive geometry [Szeged, 1991], 1994, pp. 131–137 [2]).

Notes

Acknowledgements

We thank the referee for helpful suggestions that improved the paper.

References

  1. 1.
    Z. Füredi, P.A. Loeb, On the Best Constant for the Besicovitch Covering Theorem. Proc. Am. Math. Soc. 121(4), 1063–1073 (1994). MR1249875 (95b:28003)Google Scholar
  2. 2.
    L. Guibas, J. Pach, M. Sharir, Sphere-of-influence graphs in higher dimensions, in Intuitive Geometry (Szeged, 1991) 1994, pp. 131–137. MR1383618 (97a:05183)Google Scholar
  3. 3.
    F. Harary, M.S. Jacobson, M.J. Lipman, F.R. McMorris, Abstract sphere-of-influence graphs. Math. Comput. Modelling 17(11), 77–83 (1993). Graph-Theoretic Models in Computer Science, II (Las Cruces, NM, 1988–1990), p. 1236512Google Scholar
  4. 4.
    J. Klein, G. Zachmann, Point cloud surfaces using geometric proximity graphs. Comput. Graph. 28(6), 839–850 (2004)CrossRefGoogle Scholar
  5. 5.
    T.S. Michael, T. Quint, Sphere of influence graphs: edge density and clique size. Math. Comput. Model. 20(7), 19–24 (1994). MR1299482MathSciNetCrossRefGoogle Scholar
  6. 6.
    J.M. Sullivan, Sphere packings give an explicit bound for the Besicovitch covering theorem. J. Geom. Anal. 4(2), 219–231 (1994). MR1277507MathSciNetCrossRefGoogle Scholar
  7. 7.
    G.T. Toussaint, The sphere of influence graph: theory and applications. Int. J. Inf. Technol. Comput. Sci. 14(2), 37–42 (2014)Google Scholar
  8. 8.
    G.T. Toussaint, A graph-theoretical primal sketch. Mach. Intell. Pattern Recognit. 6, 229–260 (1988). A Computational Geometric Approach to the Analysis of Form, MR993994Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Márton Naszódi
    • 1
  • János Pach
    • 2
  • Konrad Swanepoel
    • 3
  1. 1.Department of GeometryLorand Eötvös UniversityBudapestHungary
  2. 2.EPFL Lausanne and Rényi InstituteBudapestHungary
  3. 3.Department of MathematicsLondon School of Economics and Political ScienceLondonUK

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