Geodesic
Planar locally minimal trees with boundaries on a circle
Sbornik. Mathematics, Tome 215 (2024) no. 5, pp. 658-666 Cet article a éte moissonné depuis la source Math-Net.Ru

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A planar tree has a convex minimal realization if it is planar equivalent to a locally minimal tree whose boundary is the set of vertices of a convex polygon. If this polygon is inscribed in a circle, then the tree is said to have a circular minimal realization. We construct a wide class of planar trees that have convex minimal realizations but do not have circular ones. Bibliography: 9 titles.
Keywords: full Steiner trees, Steiner minimal trees, Steiner problem, locally minimal trees, twisting number of a full planar Steiner tree. 
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I. N. Mikhailov. Planar locally minimal trees with boundaries on a circle. Sbornik. Mathematics, Tome 215 (2024) no. 5, pp. 658-666. http://geodesic.mathdoc.fr/item/SM_2024_215_5_a3/

[1] A. O. Ivanov and A. A. Tuzhilin, Theory of extremal networks, Moscow–Izhevsk, Institute for Computer Studies, 2003, 424 pp. (Russian)

[2] A. O. Ivanov and A. A. Tuzhilin, Minimal networks. The Steiner problem and its generalizations, CRC Press, Boca Raton, FL, 1994, xviii+414 pp. | MR | Zbl

[3] M. R. Garey, R. L. Graham and D. S. Johnson, “The complexity of computing Steiner minimal trees”, SIAM J. Appl. Math., 32:4 (1977), 835–859 | DOI | MR | Zbl

[4] A. O. Ivanov and A. A. Tuzhilin, “Geometry of minimal networks and the one-dimensional Plateau problem”, Russian Math. Surveys, 47:2 (1992), 59–131 | DOI | MR | Zbl

[5] A. O. Ivanov and A. A. Tuzhilin, “The Steiner problem in the plane or in plane minimal nets”, Math. USSR-Sb., 74:2 (1993), 555–582 | DOI | MR | Zbl

[6] A. O. Ivanov, I. V. Iskhakov and A. A. Tuzhilin, “Minimal trees for regular polygons: linear parquets realization”, Moscow Univ. Math. Bull., 48:6 (1993), 46–48 | MR | Zbl

[7] A. O. Ivanov and A. A. Tuzhilin, “On minimal binary trees with a regular boundary”, Russian Math. Surveys, 51:1 (1996), 144–145 | DOI | MR | Zbl

[8] A. A. Tuzhilin, “Complete classification of locally minimal binary trees with regular boundary. The case of skeletons”, Fundam. Prikl. Mat., 2:2 (1996), 511–562 (Russian) | MR | Zbl

[9] A. O. Ivanov and A. A. Tuzhilin, “Non-trivial example of a boundary set in generalized Steiner problem constructed with the help of computer geometry and visualization”, Computer Graphics Geometry, 6:1 (2004), 75–99