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Discrete & Computational Geometry

, Volume 7, Issue 1, pp 77–86 | Cite as

The Steiner ratio conjecture for cocircular points

  • J. H. Rubinstein
  • D. A. Thomas
Article

Abstract

A Steiner minimal treeS is a network of shortest possible length connecting a set ofn points in the plane. LetT be a shortest tree connecting then points but with vertices only at these points.T is called a minimal spanning tree. The Steiner ratio conjecture is that the length ofS divided by the length ofT is at least √3/2. In this paper we use a variational approach to show that if then points lie on a circle, then the Steiner ratio conjecture holds.

Keywords

Minimal Span Tree Discrete Comput Geom Steiner Tree Steiner Point Longe Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    F. R. K. Chung and R. L. Graham, A new bound for Euclidean Steiner minimal trees,Ann. N. Y. Acad. Sci. 440 (1985), 328–346.MathSciNetCrossRefGoogle Scholar
  2. 2.
    D. Z. Du, F. K. Hwang, and E. N. Yao, The Steiner ratio conjecture is true for five points,J. Combin. Theory Ser. A 38 (1985), 230–240.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Z. Du, E. N. Yao, and F. K. Hwang, A short proof of a result of Pollak on Steiner minimal trees,J. Combin. Theory Ser. A 32 (1982), 396–400.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    E. N. Gilbert and H. O. Pollak, Steiner minimal trees,SIAM J. Appl. Math. 16 (1968), 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. O. Pollak, Some remarks on the Steiner problem,J. Combin. Theory Ser. A 24 (1978), 278–295.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. H. Rubinstein and D. A. Thomas, A variational approach to the Steiner network problem,Ann. Oper. Res., to appear.Google Scholar
  7. 7.
    J. H. Rubinstein and D. A. Thomas, The Steiner ratio conjecture for six points,J. Combin. Theory Ser. A, to appear.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • J. H. Rubinstein
    • 1
    • 2
  • D. A. Thomas
    • 2
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsUniversity of MelbourneParkvilleAustralia

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