Geodesic
Reduction to general position of a~mapping of a~one-dimensional polyhedron, depending continuously on a~parameter
Izvestiya. Mathematics , Tome 27 (1986) no. 2, pp. 359-389

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This paper is devoted to a proof of the fact that by refining the triangulation of a one-dimensional polyhedron, one can approximate a given mapping of that polyhedron into $\mathbf R^k$ by a piecewise linear mapping having no more than a zero-dimensional violation of general position; and that all this can be carried out continuously with respect to a parameter running through a strongly paracompact space. Spaces of triangulations of one-dimensional simplexes are also investigated, and the structure of spaces of semilinear mappings of a one-dimensional polyhedron into Euclidean space is considered. Figures: 6. Bibliography: 6 titles. 
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     author = {S. I. Yablokova},
     title = {Reduction to general position of a~mapping of a~one-dimensional polyhedron, depending continuously on a~parameter},
     journal = {Izvestiya. Mathematics },
     pages = {359--389},
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     volume = {27},
     number = {2},
     year = {1986},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1986_27_2_a7/}
}
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S. I. Yablokova. Reduction to general position of a~mapping of a~one-dimensional polyhedron, depending continuously on a~parameter. Izvestiya. Mathematics , Tome 27 (1986) no. 2, pp. 359-389. http://geodesic.mathdoc.fr/item/IM2_1986_27_2_a7/