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Mat. Sb., 2006, Volume 197, Number 9, Pages 55–90 (Mi msb1463)  

This article is cited in 5 scientific papers (total in 5 papers)

Uniqueness of Steiner minimal trees on boundaries in general position

A. O. Ivanov, A. A. Tuzhilin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The following result is proved: there exists an open dense subset $U$ of $\mathbb R^{2n}$ such that each $P\in U$ (regarded as an enumerated subset of the standard Euclidean plane $\mathbb R^2$) is spanned by a unique Steiner minimal tree, that is, a shortest non-degenerate network. Several interesting consequences are also obtained: in particular, it is proved that each planar Steiner tree is planar equivalent to a Steiner minimal tree.
Bibliography: 11 titles.

DOI: https://doi.org/10.4213/sm1463

Full text: PDF file (813 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:9, 1309–1340

Bibliographic databases:

UDC: 514.774.8+519.176
MSC: Primary 7M15; Secondary 05C05
Received: 05.12.2005

Citation: A. O. Ivanov, A. A. Tuzhilin, “Uniqueness of Steiner minimal trees on boundaries in general position”, Mat. Sb., 197:9 (2006), 55–90; Sb. Math., 197:9 (2006), 1309–1340

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Oblakov K.I., “Non-existence of distinct codirected locally minimal trees on a plane”, Moscow Univ. Math. Bull., 64:2 (2009), 62–66  crossref  mathscinet  zmath  elib  elib
    2. H. Edelsbrunner, N. P. Strelkova, “On the configuration space of Steiner minimal trees”, Russian Math. Surveys, 67:6 (2012), 1167–1168  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. H. Edelsbrunner, A. Ivanov, R. Karasev, “Current Open Problems in Discrete and Computational Geometry”, Model. i analiz inform. sistem, 19:5 (2012), 5–17  mathnet
    4. Zachos A.N., “An Evolutionary Structure of Convex Pentagons on a C-2 Complete Surface and a Creation Principle of Some Weighted Dendrites of Order Three”, J. Convex Anal., 20:4 (2013), 1043–1073  mathscinet  zmath  isi  elib
    5. Ivanov A.O. Tuzhilin A.A., “Minimal Networks: a Review”, Advances in Dynamical Systems and Control, Studies in Systems Decision and Control, 69, ed. Sadovnichiy V. Zgurovsky M., Springer Int Publishing Ag, 2016, 43–80  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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