, Volume 7, Issue 1–6, pp 193–218 | Cite as

Graham's problem on shortest networks for points on a circle

  • J. H. Rubinstein
  • D. A. Thomas


Suppose a configurationX consists ofn points lying on a circle of radiusr. If at most one of the edges joining neighboring points has length strictly greater thanr, then the Steiner treeS consists of all these edges with a longest edge removed. In order to showS is, in fact, just the minimal spanning treeT, a variational approach is used to show the Steiner ratio for this configuration is at least one and equals one only ifS andT coincide. The variational approach greatly reduces the number of possible Steiner trees that need to be considered.

Key words

Graham's conjecture Steiner trees Spanning trees Variational approach Cocircular points 


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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • J. H. Rubinstein
    • 1
  • D. A. Thomas
    • 2
  1. 1.Institute of Advanced Study, and Mathematics DepartmentUniversity of MelbourneParkvilleAustralia
  2. 2.University of MelbourneParkvilleAustralia

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