Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

## 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 to check access.

1. 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 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. 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)

7. 7.

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

8. 8.

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

9. 9.

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

10. 10.

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

11. 11.

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

12. 12.

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

13. 13.

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

## Acknowledgments

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

## Author information

Correspondence to Chaya Keller.