Geodesic
The geometry of minimal networks with a~given topology and a~fixed boundary
Izvestiya. Mathematics , Tome 61 (1997) no. 6, pp. 1231-1263

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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. 
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     author = {A. O. Ivanov and A. A. Tuzhilin},
     title = {The geometry of minimal networks with a~given topology and a~fixed boundary},
     journal = {Izvestiya. Mathematics },
     pages = {1231--1263},
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     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_6_a4/}
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A. O. Ivanov; A. A. Tuzhilin. The geometry of minimal networks with a~given topology and a~fixed boundary. Izvestiya. Mathematics , Tome 61 (1997) no. 6, pp. 1231-1263. http://geodesic.mathdoc.fr/item/IM2_1997_61_6_a4/