Skip to main content
SpringerLink
Log in

Straight Line Triangle Representations

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

Abstract

A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size \(\pi \) in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them. The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. The labels are considered in a cyclic structure, such that \((i-1)\) and \((i+1)\) are always well defined.

  2. Almost 4-regular graphs are Laman graphs. The number of edges is twice the number of vertices minus three and this is an upper bound for each subset of the vertices.

References

  1. Aerts, N., Felsner, S.: Henneberg steps for triangle representations. In: Proceedings EuroComb 2013, CRM, vol. 16, pp. 503–509. Scuola Normale Superiore, Pisa (2013)

  2. Aerts, N., Felsner, S.: Straight-line triangle representations via Schnyder labelings. J. Graph Alg. Appl. 19, 467–505 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alam, M.J., Fowler, J., Kobourov, S.G.: Outerplanar graphs with proper touching triangle representations (unpublished)

  4. Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18, 19–37 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Felsner, S.: Geodesic embeddings and planar graphs. Order 20, 135–150 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Felsner, S.: Lattice structures from planar graphs. Electron. J. Comb. 11(R15), 24p (2004)

    MathSciNet  MATH  Google Scholar 

  7. Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Comb. 15, 41p (2008)

    MathSciNet  MATH  Google Scholar 

  8. Felsner, S., Zickfeld, F.: Schnyder woods and orthogonal surfaces. Discrete Comput. Geom. 40, 103–126 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fowler, J.J.: Strongly-connected outerplanar graphs with proper touching triangle representations. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 8242, pp. 156–161. Springer, Heidelberg (2013)

  10. de Fraysseix, H., de Mendez, P.O.: Barycentric systems and stretchability. Discrete Appl. Math. 155, 1079–1095 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Comb. Probab. Comput. 3(02), 233–246 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gansner, E.R., Hu, Y., Kobourov, S.G.: On touching triangle graphs. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 6502, pp. 250–261. Springer, Heidelberg (2010)

  13. Gonçalves, D., Lévêque, B., Pinlou, A.: Triangle contact representations and duality. Discrete Comput. Geom. 48(1), 239–254 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Haas, R., Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Souvaine, D.L., Streinu, I., Whiteley, W.: Planar minimally rigid graphs and pseudo-triangulations. Comput. Geom. 31, 31–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kenyon, R., Sheffield, S.: Dimers, tilings and trees. J. Comb. Theory Ser. B 92, 295–317 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kobourov, S.G., Mondal, D., Nishat, R.I.: Touching triangle representations for 3-connected planar graphs. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 7704, pp. 199–210. Springer, New York (2012)

  17. Lovász, L.: Geometric representations of graphs (Draft version December 11, 2009). http://www.cs.elte.hu/~lovasz/geomrep.pdf

  18. Miller, E.: Planar graphs as minimal resolutions of trivariate monomial ideals. Doc. Math. 7, 43–90 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the ACM–SIAM Symposium on Discrete Algorithms, pp. 138–148 (1990)

  21. Tutte, W.T.: How to draw a graph. Proc. Lond. Math. Soc. 13, 743–767 (1963)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Partially supported by DFG Grant FE-340/7-2 and ESF EuroGIGA project GraDR.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Nieke Aerts & Stefan Felsner

Authors
  1. Nieke Aerts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Felsner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nieke Aerts.

Additional information

Editor in Charge: János Pach

An extended abstract of this paper was presented at the 21st International Symposium on Graph Drawing (GD2013).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aerts, N., Felsner, S. Straight Line Triangle Representations. Discrete Comput Geom 57, 257–280 (2017). https://doi.org/10.1007/s00454-016-9850-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-016-9850-y

Keywords

Mathematics Subject Classification

Search

Navigation