Skip to main content
Log in

Reconstruction of the Geometric Structure of a Set of Points in the Plane from Its Geometric Tree Graph

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

Let P be a finite set of points in general position in the plane. The structure of the complete graph K(P) as a geometric graph includes, for any pair [ab], [cd] of vertex-disjoint edges, the information whether they cross or not. The simple (i.e., non-crossing) spanning trees (SSTs) of K(P) are the vertices of the so-called Geometric Tree Graph of P, G(P). Two such vertices are adjacent in G(P) if they differ in exactly two edges, i.e., if one can be obtained from the other by deleting an edge and adding another edge. In this paper we show how to reconstruct from G(P) (regarded as an abstract graph) the structure of K(P) as a geometric graph. We first identify within G(P) the vertices that correspond to spanning stars. Then we regard each star S(z) with center z as the representative in G(P) of the vertex z of K(P). (This correspondence is determined only up to an automorphism of K(P) as a geometric graph.) Finally we determine for any four distinct stars S(a), S(b), S(c),  and S(d), by looking at their relative positions in G(P), whether the corresponding segments cross.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. For sake of clarity, we use here and in the sequel the notation \([T,T']\) for edges of G(P), like is commonly used for geometric graphs, although G(P) is treated as an abstract graph.

References

  1. Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1), 21–46 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bondy, J.A., Hemminger, R.L.: Graph reconstruction—a survey. J. Graph Theory 1, 227–268 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cummins, R.L.: Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory 13(1), 82–90 (1966)

    Article  MathSciNet  Google Scholar 

  4. Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)

    MathSciNet  MATH  Google Scholar 

  5. Hernando, M.C.: Complejidad de Estructuras Geométricas y Combinatorias, Ph.D. Thesis, Universitat Politéctnica de Catalunya, 1999 (in Spanish). http://www.tdx.cat/TDX-0402108-120036/

  6. Hernando, M.C., Hurtado, F., Márquez, A., Mora, M., Noy, M.: Geometric tree graphs of points in convex position. Discrete Appl. Math. 93(1), 51–66 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Holzmann, C.A., Harary, F.: On the tree graph of a matroid. SIAM J. Appl. Math. 22, 187–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kelly, P.J.: A congruence theorem for trees. Pac. J. Math. 7, 961–968 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, G.: On connectivities of tree graphs. J. Graph Theory 12, 453–459 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ramachandran, S.: Graph reconstruction—some new developments. AKCE J. Graphs Comb. 1(1), 51–61 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Sedláček, J.: The reconstruction of a connected graph from its spanning trees. Mat. Časopis Sloven. Akad. Vied. 24, 307–314 (1974)

    MathSciNet  MATH  Google Scholar 

  12. Ulam, S.M.: A Collection of Mathematical Problems. Wiley, New York (1960)

    MATH  Google Scholar 

  13. Urrutia-Galicia, V.: Algunas Propiedades de Gráficas Geométricas, Ph.D. Thesis, Universidad Autonóma Metropolitana Unidad Iztapalapa, México D.F. (2001) (in Spanish)

Download references

Acknowledgments

The work of Chaya Keller was partially supported by the Hoffman Leadership and Responsibility Program at the Hebrew University.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chaya Keller.

Additional information

Editor in Charge: János Pach

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keller, C., Perles, M.A. Reconstruction of the Geometric Structure of a Set of Points in the Plane from Its Geometric Tree Graph. Discrete Comput Geom 55, 610–637 (2016). https://doi.org/10.1007/s00454-015-9750-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-015-9750-6

Keywords

Navigation