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Izv. RAN. Ser. Mat., 1997, Volume 61, Issue 6, Pages 119–152 (Mi izv168)  

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

The geometry of minimal networks with a given topology and a fixed boundary

A. O. Ivanov, A. A. Tuzhilin

M. V. Lomonosov Moscow State University

Abstract: In this paper we study the structure of the set $\mathcal M_G(\varphi)$ of all locally minimal plane networks with a fixed topology $G$ and a fixed boundary $\varphi$. It is shown that if this set is non-empty, then it is a $k$-dimensional convex body in the configuration space $\mathbb R^N$ of the movable vertices of the network, where $k$ is the cyclomatic number for the movable subgraph in $G$.
In particular, all the networks in $\mathcal M_G(\varphi)$ are parallel, have the same length, and can be deformed into one another in the class of locally minimal networks of the same type and with the same boundary. Moreover, we describe how two networks belonging to $\mathcal M_G(\varphi)$ can be distinguished.

DOI: https://doi.org/10.4213/im168

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English version:
Izvestiya: Mathematics, 1997, 61:6, 1231–1263

Bibliographic databases:

MSC: 05C05, 05C35, 68R10, 90C35
Received: 01.03.1996

Citation: A. O. Ivanov, A. A. Tuzhilin, “The geometry of minimal networks with a given topology and a fixed boundary”, Izv. RAN. Ser. Mat., 61:6 (1997), 119–152; Izv. Math., 61:6 (1997), 1231–1263

Citation in format AMSBIB
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\paper The geometry of minimal networks with a~given topology and a~fixed boundary
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. O. Ivanov, A. A. Tuzhilin, “The space of parallel linear networks with a fixed boundary”, Izv. Math., 63:5 (1999), 923–962  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. G. A. Karpunin, “An analogue of Morse theory for planar linear networks and the generalized Steiner problem”, Sb. Math., 191:2 (2000), 209–233  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. G. A. Karpunin, “Minimal Networks on the Regular $n$-Dimensional Simplex”, Math. Notes, 69:6 (2001), 780–789  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. O. Ivanov, A. A. Tuzhilin, “Branching geodesics in normed spaces”, Izv. Math., 66:5 (2002), 905–948  mathnet  crossref  crossref  mathscinet  zmath
    5. N. P. Strelkova, “Zamknutye lokalno minimalnye seti na poverkhnostyakh vypuklykh mnogogrannikov”, Model. i analiz inform. sistem, 20:5 (2013), 117–147  mathnet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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