Skip to main content
SpringerLink
Log in

Geometry and topology of local minimal 2-trees

  • Published:
Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society Aims and scope Submit manuscript

Abstract

The aim of this paper is to get an effective restriction on the topologies of minimal 2-trees in terms of twisting numbers of the trees and the convexity levels number of the trees boundaries.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. B. Kruskal, On the shortest spanning subtree of a graph and traveling salesman problem. Proc. Amer. Math. Soc., 1956, vol. 7, pp. 48–50.

    Google Scholar 

  2. R. C. Prim, Shortest connecting networks and some generalizations. BSTJ, 1957, vol. 36, pp. 1389–1401.

    Google Scholar 

  3. E. W. Dijkstra, A note on two problems with connection with graphs. Numer. Math., 1959, vol. 1, no. 5, pp. 269–271.

    Google Scholar 

  4. Z. A. Melzak, On the problem of Steiner. Canad. Math. Bull., 1960, vol. 4, pp. 143–148.

    Google Scholar 

  5. E. N. Gilbert and H. O. Pollak, Steiner minimal trees. SIAM J. Appl. Math., 1968, vol. 16, no. 1, pp. 1–29.

    Google Scholar 

  6. D. Z. Du and F. K. Hwang, A New Bound for the Steiner Ratio. Trans. Amer. Math. Soc. 1983, vol. 278, no. 1, pp. 137–148.

    Google Scholar 

  7. D. Z. Du, F. K. Hwang and J. F. Weng, Steiner minimal trees for points on a zig-zag lines. Trans. Amer. Math. Soc. 1983, vol. 278, no. 1, pp. 149–156.

    Google Scholar 

  8. D. Z. Du, F. K. Hwang and S. C. Chao, Steiner minimal trees for points on a circle. Proc. Amer. Math. Soc., 1985, vol. 95, no. 4, pp. 613–618.

    Google Scholar 

  9. D. Z. Du, F. K. Hwang and E. N. Yao, The Steiner ratio conjecture is true for five points. J. Combin Thy., Ser. A38, 1985, pp. 230–240.

    Google Scholar 

  10. F. K. Hwang, A linear time algorithm for full Steiner trees. Oper. Res. Letter, 1986, vol. 5, pp. 235–237.

    Google Scholar 

  11. F. K. Hwang and J. F. Weng, Hexagonal coordinate Systems and Steiner minimal trees. Disc. Math., 1986, vol. 62, pp. 49–57.

    Google Scholar 

  12. D. Z. Du, F. K. Hwang and J. F. Weng, Steiner minimal trees for Regular Polygons. Disc. and Comp. Geometry, 1987, vol. 2, pp. 65–84.

    Google Scholar 

  13. D. Z. Du, F. K. Hwang, G. D. Song and G. T. Ting, Steiner minimal trees on sets of four points. Discr. and Comp. Geometry, 1987, vol. 2, pp. 401–414.

    Google Scholar 

  14. Du D.Z., Hwang F.K., A Proof of Gilbert-Pollak's Conjecture on the Steiner Ratio. DIMACS Technical Report, 1990, no. 90-72.

  15. Du D.Z., Hwang F.K., An approach for proving lower bounds: solution of Gilbert-Pollak's Conjecture on the Steiner Ratio. Proc of the 31st annual symp. on found. of comp. science, 1990.

  16. F. K. Hwang, D. Richards and P. Winter, The Steiners Tree Problem. Elsevier Science Publishers (to appear).

  17. E. J. Cockayne, On the Steiner problem. Canad. J. Math., 1967, vol. 10, pp. 431–450.

    Google Scholar 

  18. H. O. Pollak, Some remarks on the Steiner problem. J. Combin. Thy., Ser. A24, 1978, pp. 278–295.

    Google Scholar 

  19. R. C. Clark, Communication networks, soap films and vectors. Phys. Ed., 1981, vol. 16, pp. 32–37.

    Google Scholar 

  20. J. H. Rubinstein and D. A. Thomas, The Steiner ratio conjecture for six points. J. Combin. Thy., Ser. A58, 1989, pp. 54–77.

    Google Scholar 

  21. W. D. Smith, How to find Steiner minimal trees in Euclideand-space. Algoritmica (to appear).

  22. A. L. Edmonds, J. H. Ewing and R. S. Kulkarni, Regular tesselations of surfaces and (p, q, 2)-triangle groups. Ann. Math., 1982, vol. 116, pp. 113–132.

    Google Scholar 

  23. A. L. Edmonds, J. H. Ewing and R. S. Kulkarni, Torsion free subgroups of Fuchian groups and tesselations of surfaces. Inv. Math., 1982, vol. 69, pp. 331–346.

    Google Scholar 

  24. Emel'ichev etc., Lections on Graphs Theory. Nauka, Moscow, 1990.

  25. A. T. Fomenko, The Plateau Problem. New York, Gordon and Breach Sc. Publ., 1989.

    Google Scholar 

  26. A. T. Fomenko, Variational problems in Topology. New York, Gordon and Breach Sc. Publ., 1990.

    Google Scholar 

  27. A.T.Fomenko and A.A.Tuzhilin, Elements of geometry and topology of minimal surfaces in three-dimensional space. AMS, 1992, vol. 93, Translations of Mathematical Monographs.

  28. Z. A. Melzak, Companion to concrete mathematics. Wiley-Interscience, New York, 1973.

    Google Scholar 

  29. S. Hildebrandt and A. Tromba, Mathematics and optimal form. An imprint of Scientific American Books, Inc., New York, 1984.

    Google Scholar 

  30. M. Zacharias, Encyklopädie der Mathematischen, Wissenschaften, vol. III AB9.

  31. H. W. Kuhn, Steiner's problem revisted. In the book Studies in Optimization, ser. Studies in Math., vol. 10, Math. Assoc. Amer., edited by G. B. Dantzig and B. C. Eaves, 1975, pp. 53–70.

  32. V. Jarnik and M. Kössler, O minimalnich grafeth obeahujicich n danijch bodu. Cas. Pest. Mat. a Fys., 1934, vol. 63, pp. 223–235.

    Google Scholar 

  33. F. Preparata and M. Shamos, Computational Geometry. An introduction. New York, Springer-Verlag, 1985.

    Google Scholar 

  34. A. O. Ivanov and A. A. Tuzhilin, Solution of the Steiner Problem for convex boundaries. Uspekhi. Mat. Nauk, 1990, vol. 45, no. 2, pp. 207–208, Russian. English transl. in Russian Math. Surveys (1990)

    Google Scholar 

  35. A. O. Ivanov and A. A. Tuzhilin, The Steiner problem for convex boundaries or flat minimal networks. Matem. Sbornik, 1991, vol. 182, no. 12, pp. 1813–1844. Russian.

    Google Scholar 

  36. A. O. Ivanov and A. A. Tuzhilin, Geometry of minimal networks and one-di-mensional Plateau problem. Uspekhi Mat. Nauk, 1992, vol. 47, no. 2 (284), Russian.

    Google Scholar 

  37. A. O. Ivanov and A. A. Tuzhilin, The Steiner problem for convex boundaries, 1: general case. Advances in Soviet Mathematics, 1993, vol. 15, pp. 15–92.

    Google Scholar 

  38. A. O. Ivanov and A. A. Tuzhilin, The Steiner problem for convex boundaries, 2: the regular case. Advances in Soviet Mathematics, 1993, vol. 15, pp. 93–131.

    Google Scholar 

  39. A. O. Ivanov and A. A. Tuzhilin, Minimal Networks. The Steiner Problem and Its Generalizations. N.W., Boca Raton, Florida, CRC Press, 1994.

    Google Scholar 

  40. A. O. Ivanov, I. Iskhakov, and A. A. Tuzhilin, Minimal networks spanning regularn-gons: linear tilings realization. Vestnik Moskov. Univ. Ser. Mat., 1993. English transl. in Moscow Univ. Math. Bull.

  41. A. O. Ivanov, I. V. Ptitsina, and A. A. Tuzhilin, Classification of closed minimal networks on flat two-dimensional tori. Matem. Sbornik, 1992, vol. 183, no. 12, pp. 3–44 (Russian).

    Google Scholar 

  42. I. V. Shklyanko, The one dimensional Plateau problem on surfaces. Vestnik Moskov. Univ. Ser. Mat., 1989, no. 3, pp. 8–11 (Russian). English transl. in Moscow Univ. Math. Bull. (1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Differential Geometry/Appl Faculty of Mechanics/Mat, Moscow State University, 119899, Moscow, Russia

    A. O. Ivanov & A. A. Tuzhilin

Authors
  1. A. O. Ivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Tuzhilin
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Ivanov, A.O., Tuzhilin, A.A. Geometry and topology of local minimal 2-trees. Bol. Soc. Bras. Mat 28, 103–139 (1997). https://doi.org/10.1007/BF01235991

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01235991

Keywords

Search

Navigation