Geodesic
An analogue of Morse theory for planar linear networks and the generalized Steiner problem
Sbornik. Mathematics, Tome 191 (2000) no. 2, pp. 209-233 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {An analogue of {Morse} theory for planar linear networks and the~generalized {Steiner} problem},
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     volume = {191},
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     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_2_a2/}
}
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G. A. Karpunin. An analogue of Morse theory for planar linear networks and the generalized Steiner problem. Sbornik. Mathematics, Tome 191 (2000) no. 2, pp. 209-233. http://geodesic.mathdoc.fr/item/SM_2000_191_2_a2/

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