Degree-five Steiner points cannot reduce network costs for planar sets

@article{Rubinstein1992DegreefiveSP,
  title={Degree-five Steiner points cannot reduce network costs for planar sets},
  author={J. Rubinstein and Doreen A. Thomas and J. Weng},
  journal={Networks},
  year={1992},
  volume={22},
  pages={531-537}
}
We show that a degree-five Steiner point can never appear in a least-cost planar network; that is we show that a degree-five Steiner point actally increases the cost. 

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References

SHOWING 1-6 OF 6 REFERENCES
An algorithm for the steiner problem in the euclidean plane
  • P. Winter
  • Mathematics, Computer Science
  • Networks
  • 1985
TLDR
An algorithm for the exact solution of the Steiner problem in the Euclidean plane is presented, which appears to be considerably faster than any other existing algorithm. Expand
Exact Computation of Steiner Minimal Trees in the Plane
TLDR
Improvements to an algorithm of Winter (1981) are presented, which enable us to solve all 17-point problems and an estimated 80% of all randomly generated problems with n⩽ 30. Expand
A variational approach to the Steiner network problem
TLDR
The variational approach was used by us to solve other cases of the ratio conjecture and solve Graham's problem to decide when the Steiner network is the same as the minimal spanning tree, for points on a circle and on any convex polygon, again using the variational method. Expand
An O(n log n) heuristic for steiner minimal tree problems on the euclidean metric
TLDR
An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented and is shown to be as good as the previous O( n4) algorithm in achieving reductions in the ratio SMT/MST of the given vertex set. Expand
Euclidean Constructibility in Graph-Minimization Problems
1. Let b1, * * *, bN be any set of distinct points in the plane. By a tree U on the vertices b1, , bv we mean any set consisting of some of the (2) closed straight segments bibj with the propertyExpand