Geodesic
The Structure of Minimal Steiner Trees in the Neighborhoods of the Lunes of Their Edges
Matematičeskie zametki, Tome 91 (2012) no. 3, pp. 353-370.

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We give a complete description of small neighborhoods of the closures of lunes of the edges of Steiner minimal trees (Theorem 1.1); to this end, we prove a generalization of a stabilization theorem for embedded locally minimal trees [1]; the case of two such disjoint trees is considered (Theorem 2.2).
Keywords: Steiner minimal tree, locally minimal tree, lune of an edge of a tree, linear graph, shortest tree. 
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A. O. Ivanov; O. A. S'edina; A. A. Tuzhilin. The Structure of Minimal Steiner Trees in the Neighborhoods of the Lunes of Their Edges. Matematičeskie zametki, Tome 91 (2012) no. 3, pp. 353-370. http://geodesic.mathdoc.fr/item/MZM_2012_91_3_a3/

[1] A. O. Ivanov, A. A. Tuzhilin, “Stabilizatsiya lokalno minimalnykh derevev”, Matem. zametki, 86:4 (2009), 512–524 | MR | Zbl

[2] A. O. Ivanov, A. A. Tuzhilin, Teoriya ekstremalnykh setei, Izd-vo IKI, M., 2003

[3] M. R. Garey, R. L. Graham, D. S. Johnson, “Some NP-complete geometric problems”, Eighth Annual ACM Symposium on Theory of Computing (Hershey, Pa., 1976), Assoc. Comput. Mach.,, New York, 1976, 10–22 | MR | Zbl

[4] E. N. Gilbert, H. O. Pollak, “Steiner minimal trees”, SIAM J. Appl. Math., 16:1 (1968), 1–29 | DOI | MR | Zbl