Skip to main content
SpringerLink
Log in

On the Structure of Graphs with Low Obstacle Number

  • Proceedings Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The obstacle number of a graph G is the smallest number of polygonal obstacles in the plane with the property that the vertices of G can be represented by distinct points such that two of them see each other if and only if the corresponding vertices are joined by an edge. We list three small graphs that require more than one obstacle. Using extremal graph theoretic tools developed by Prömel, Steger, Bollobás, Thomason, and others, we deduce that for any fixed integer h, the total number of graphs on n vertices with obstacle number at most h is at most \({2^{o(n^2)}}\). This implies that there are bipartite graphs with arbitrarily large obstacle number, which answers a question of Alpert et al. (Discret Comput Geom doi:10.1007/s00454-009-9233-8, 2009).

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. Alpert, H., Koch, C., Laison, J.: Obstacle numbers of graphs. Discret. Computat. Geom. (2009). doi:10.1007/s00454-009-9233-8 . http://dx.doi.org/10.1007/s00454-009-9233-8 . Published at http://www.springerlink.com/content/45038g67t22463g5 (viewed on 12/26/09), 27 p

  2. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. Algorithms and Applications (2nd edn.). Springer, Berlin (2000)

  3. Bollobás, B., Thomason, A.: Hereditary and monotone properties of graphs. In: Graham, R. L., Nešetřil, J. (eds.) The mathematics of Paul Erdős vol. 2, Algorithms and Combinatorics 14, pp. 70–78. Springer, Berlin (1997)

  4. Erdős P., Frankl P., Rödl V.: The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graph Comb. 2, 113–121 (1986)

    Article  Google Scholar 

  5. Erdős, P., Kleitman, D.J., Rothschild, B.L.: Asymptotic enumeration of K n -free graphs. In: Colloq. int. Teorie comb., Roma, Tomo II, pp. 19–27 (1976)

  6. Erdös P., Szekeres G.: A combinatorial problem in geometry. Composit. Math. 2, 463–470 (1935)

    MATH  Google Scholar 

  7. Foldes S., Hammer P.L.: Split graphs having Dilworth number 2. Can. J. Math. (Journal Canadien de Mathematiques) 29(3), 666–672 (1977)

    MATH  MathSciNet  Google Scholar 

  8. Ghosh, S.K.: Visibility algorithms in the plane. Cambridge University Press, Cambridge (2007). doi:10.1017/CBO9780511543340

  9. O’Rourke, J.: Visibility. In: Handbook of discrete and computational geometry, CRC Press Ser. Discret. Math. Appl., pp. 467–479. CRC, Boca Raton (1997)

  10. O’Rourke, J.: Open problems in the combinatorics of visibility and illumination. In: Advances in discrete and computational geometry (South Hadley, MA, 1996), Contemp. Math., vol. 223, pp. 237–243. Amer. Math. Soc., Providence (1999)

  11. Pach, J., Sarioz, D.: Small (2, s)-colorable graphs without 1-obstacle representations (2010). http://arxiv.org/abs/1012.5907

  12. Prömel H.J., Steger A.: Excluding induced subgraphs: quadrilaterals. Random Struct. Algorithms 2(1), 55–71 (1991)

    Article  MATH  Google Scholar 

  13. Prömel H.J., Steger A.: Excluding induced subgraphs. III. A general asymptotic. Random Struct. Algorithms 3(1), 19–31 (1992)

    Article  MATH  Google Scholar 

  14. Prömel H.J., Steger A.: Excluding induced subgraphs. II. Extremal graphs. Discrete Applied Mathematics 44, 283–294 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Tyškevič, R. I., Černjak, A. A.: Canonical decomposition of a graph determined by the degrees of its vertices. Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 5(5), 14–26, 138 (1979)

    Google Scholar 

  16. Urrutia, J.: Art gallery and illumination problems. In: Handbook of computational geometry, pp. 973–1027. North-Holland, Amsterdam (2000). doi:10.1016/B978-044482537-7/50023-1

Download references

Author information

Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    János Pach

  2. The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    János Pach & Deniz Sarıöz

Authors
  1. János Pach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deniz Sarıöz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to János Pach.

Additional information

Research supported by NSA grant 47149-0001, NSF grant CCF-08-30272, Swiss National Science Foundation grant 200021-125287/1, and by the Bernoulli Center at EPFL.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pach, J., Sarıöz, D. On the Structure of Graphs with Low Obstacle Number. Graphs and Combinatorics 27, 465–473 (2011). https://doi.org/10.1007/s00373-011-1027-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-011-1027-0

Keywords

Mathematics Subject Classification (1991)

Search

Navigation