Mathematical Notes

, Volume 91, Issue 3–4, pp 339–353 | Cite as

The structure of minimal Steiner trees in the neighborhoods of the lunes of their edges

Article

Abstract

We give a complete description of small neighborhoods of the closures of lunes of the edges of Steiner minimal trees (Theorem 1.1); to this end, we prove a generalization of a stabilization theorem for embedded locally minimal trees [1]; the case of two such disjoint trees is considered (Theorem 2.2).

Keywords

Steiner minimal tree locally minimal tree lune of an edge of a tree linear graph shortest tree 

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References

  1. 1.
    A. O. Ivanov and A. A. Tuzhilin, “Stabilization of locally minimal trees,” Mat. Zametki 86(4), 512–524 (2009) [Math. Notes 86 (4), 483–492 (2009)].MathSciNetGoogle Scholar
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    A. O. Ivanov and A. A. Tuzhilin, The Theory of Extremal Nets (Izd. IKI, Moscow, 2003)] [in Russian].Google Scholar
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    M. R. Garey, R. L. Graham, and D. S. Johnson, “Some NP-complete geometric problems,” in Eighth Annual ACM Symposium on Theory of Computing, Hershey, Pa., 1976 (Assoc. Comput. Mach., New York, 1976), pp. 10–22.CrossRefGoogle Scholar
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    E. N. Gilbert and H. O. Pollak, “Steiner minimal trees,” SIAM J. Appl.Math. 16(1), 1–29 (1968).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • A. O. Ivanov
    • 1
  • O. A. S″edina
    • 2
  • A. A. Tuzhilin
    • 1
  1. 1.Moscow State UniversityYaroslavl State UniversityMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia

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