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Mat. Sb., 2000, Volume 191, Number 2, Pages 64–90 (Mi msb453)  

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

An analogue of Morse theory for planar linear networks and the generalized Steiner problem

G. A. Karpunin

M. V. Lomonosov Moscow State University

Abstract: A study is made of the generalized Steiner problem: the problem of finding all the locally minimal networks spanning a given boundary set (terminal set). It is proposed to solve this problem by using an analogue of Morse theory developed here for planar linear networks. The space $\mathscr K$ of all planar linear networks spanning a given boundary set is constructed. The concept of a critical point and its index is defined for the length function $\ell$ of a planar linear network. It is shown that locally minimal networks are local minima of $\ell$ on $\mathscr K$ and are critical points of index 1. The theorem is proved that the sum of the indices of all the critical points is equal to $\chi(\mathscr K)=1$. This theorem is used to find estimates for the number of locally minimal networks spanning a given boundary set.

DOI: https://doi.org/10.4213/sm453

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English version:
Sbornik: Mathematics, 2000, 191:2, 209–233

Bibliographic databases:

UDC: 514.772+519.711.72+519.711.74
MSC: Primary 05C35, 05C05; Secondary 58E05, 90C35
Received: 16.03.1999

Citation: G. A. Karpunin, “An analogue of Morse theory for planar linear networks and the generalized Steiner problem”, Mat. Sb., 191:2 (2000), 64–90; Sb. Math., 191:2 (2000), 209–233

Citation in format AMSBIB
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\pages 64--90
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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. 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
    2. Ivanov, AO, “Extreme networks”, Acta Applicandae Mathematicae, 66:3 (2001), 251  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. D. P. Il'yutko, “Branching extremals of the functional of $\lambda$-normed length”, Sb. Math., 197:5 (2006), 705–726  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. E. I. Stepanova, “Bifurcations of Steiner minimal trees and minimal fillings for non-convex four-point boundaries and Steiner subratio for the Euclidean plane”, Moscow University Mathematics Bulletin, 71:2 (2016), 79–81  mathnet  crossref  mathscinet  isi
    5. D. P. Ilyutko, I. M. Nikonov, “Extremal networks in $\lambda$-geometry, where $\lambda=3,4,6$”, Sb. Math., 208:4 (2017), 479–509  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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