Geodesic
Linear nets and convex polyhedra
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 3, Tome 252 (1998), pp. 52-61

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It is proved that the set $[G,\varphi]_\Gamma$ of immersed linear networks in $\mathbb R^N$ which are parallel to a given immersed linear network $\Gamma\colon G\to\mathbb R^N$ and have the same boundary $\varphi$ as $\Gamma$, can be configuration space of movable vertices of the graph $G$. Also, the dimension of the space $[G,\varphi]_\Gamma$ is calculated, and the number of faces is estimated. As an application, the space of all local minimal and weighted local minimal networks in $\mathbb R^N$ with fixed topology and boundary is described. 
@article{ZNSL_1998_252_a5,
     author = {A. O. Ivanov and A. A. Tuzhilin},
     title = {Linear nets and convex polyhedra},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {52--61},
     publisher = {mathdoc},
     volume = {252},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_252_a5/}
}
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A. O. Ivanov; A. A. Tuzhilin. Linear nets and convex polyhedra. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 3, Tome 252 (1998), pp. 52-61. http://geodesic.mathdoc.fr/item/ZNSL_1998_252_a5/