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 Mat. Sb., 2014, Volume 205, Number 3, Pages 83–118 (Mi msb8222)

Stabilization of a locally minimal forest

A. O. Ivanovab, A. E. Mel'nikovaa, A. A. Tuzhilina

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b N. E. Bauman Moscow State Technical University

Abstract: The method of partial stabilization of locally minimal networks, which was invented by Ivanov and Tuzhilin to construct examples of shortest trees with given topology, is developed. According to this method, boundary vertices of degree $2$ are not added to all edges of the original locally minimal tree, but only to some of them. The problem of partial stabilization of locally minimal trees in a finite-dimensional Euclidean space is solved completely in the paper, that is, without any restrictions imposed on the number of edges remaining free of subdivision. A criterion for the realizability of such stabilization is established. In addition, the general problem of searching for the shortest forest connecting a finite family of boundary compact sets in an arbitrary metric space is formalized; it is shown that such forests exist for any family of compact sets if and only if for any finite subset of the ambient space there exists a shortest tree connecting it. The theory developed here allows us to establish further generalizations of the stabilization theorem both for arbitrary metric spaces and for metric spaces with some special properties.
Bibliography: 10 titles.

Keywords: metric spaces, locally minimal trees, minimal Steiner trees, shortest trees, shortest forests, stabilization of a locally minimal tree.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 11.G34.31.0053ÍØ-1410.2012.1 Russian Foundation for Basic Research 13-01-00664a

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8222

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English version:
Sbornik: Mathematics, 2014, 205:3, 387–418

Bibliographic databases:

UDC: 514.774.8+515.124.2+515.124.4+514.177.2

Citation: A. O. Ivanov, A. E. Mel'nikova, A. A. Tuzhilin, “Stabilization of a locally minimal forest”, Mat. Sb., 205:3 (2014), 83–118; Sb. Math., 205:3 (2014), 387–418

Citation in format AMSBIB
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