Graphs and Combinatorics

, Volume 27, Issue 3, pp 465–473 | Cite as

On the Structure of Graphs with Low Obstacle Number

  • János PachEmail author
  • Deniz Sarıöz
Proceedings Paper


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).


Obstacle number Visibility graph Hereditary graph property Forbidden induced subgraphs Split graphs Enumeration 

Mathematics Subject Classification (1991)

05C62 05C75 68R10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alpert, H., Koch, C., Laison, J.: Obstacle numbers of graphs. Discret. Computat. Geom. (2009). doi: 10.1007/s00454-009-9233-8 . . Published at (viewed on 12/26/09), 27 p
  2. 2.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. Algorithms and Applications (2nd edn.). Springer, Berlin (2000)Google Scholar
  3. 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)Google Scholar
  4. 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)CrossRefGoogle Scholar
  5. 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)Google Scholar
  6. 6.
    Erdös P., Szekeres G.: A combinatorial problem in geometry. Composit. Math. 2, 463–470 (1935)zbMATHGoogle Scholar
  7. 7.
    Foldes S., Hammer P.L.: Split graphs having Dilworth number 2. Can. J. Math. (Journal Canadien de Mathematiques) 29(3), 666–672 (1977)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Ghosh, S.K.: Visibility algorithms in the plane. Cambridge University Press, Cambridge (2007). doi: 10.1017/CBO9780511543340
  9. 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)Google Scholar
  10. 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)Google Scholar
  11. 11.
    Pach, J., Sarioz, D.: Small (2, s)-colorable graphs without 1-obstacle representations (2010).
  12. 12.
    Prömel H.J., Steger A.: Excluding induced subgraphs: quadrilaterals. Random Struct. Algorithms 2(1), 55–71 (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Prömel H.J., Steger A.: Excluding induced subgraphs. III. A general asymptotic. Random Struct. Algorithms 3(1), 19–31 (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Prömel H.J., Steger A.: Excluding induced subgraphs. II. Extremal graphs. Discrete Applied Mathematics 44, 283–294 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 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. 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

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.The Graduate Center of the City University of New YorkNew YorkUSA

Personalised recommendations