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Graphs with Large Obstacle Numbers

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

Motivated by questions in computer vision and sensor networks, Alpert et al. [3] introduced the following definitions. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if an only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It was shown in [3] that there exist graphs of n vertices with obstacle number at least \(\Omega(\sqrt{\log n})\). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω(n/log2 n), and (2) the total number of graphs on n vertices with bounded obstacle number is at most \(2^{o(n^2)}\). Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides.

Keywords

Computational Geometry Colorable Graph Order Type Visibility Graph Complete Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alon, N.: The number of polytopes, configurations and real matroids. Mathematika 33(1), 62–71 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aloupis, G., Ballinger, B., Collette, S., Langerman, S., Por, A., Wood, D.R.: Blocking coloured point sets. In: 26th European Workshop on Computational Geometry (EuroCG 2010), Dortmund, Germany (March 2010), arXiv:1002.0190v1 [math.CO] Google Scholar
  3. 3.
    Alpert, H., Koch, C., Laison, J.: Obstacle numbers of graphs. Discrete and Computational Geometry, 27 (December 2009), http://www.springerlink.com/content/45038g67t22463g5 (viewed on 12/26/09)
  4. 4.
    Arkin, E.M., Halperin, D., Kedem, K., Mitchell, J.S.B., Naor, N.: Arrangements of segments that share endpoints: single face results. Discrete Comput. Geom. 13(3-4), 257–270 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. Algorithms and Applications, 2nd edn. Springer, Berlin (2000)zbMATHGoogle Scholar
  6. 6.
    Bollobás, B., Thomason, A.: Hereditary and monotone properties of graphs. In: Graham, R.L., Nešetřil, J. (eds.) The Mathematics of Paul Erdős. Algorithms and Combinatorics 14, vol. 2, pp. 70–78. Springer, Berlin (1997)Google Scholar
  7. 7.
    Dumitrescu, A., Pach, J., Tóth, G.: A note on blocking visibility between points. Geombinatorics 19(1), 67–73 (2009)MathSciNetGoogle Scholar
  8. 8.
    Erdős, P., Hajnal, A.: Ramsey-type theorems. Discrete Appl. Math. 25(1-2), 37–52 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erdős, P.: Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53, 292–294 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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 and Combinatorics 2, 113–121 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio. Math. 2, 463–470 (1935)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Foldes, S., Hammer, P.L.: Split graphs having Dilworth number 2. Canadian Journal of Mathematics - Journal Canadien de Mathematiques 29(3), 666–672 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fox, J., Pach, J.: Erdős–Hajnal-type results on intersection patterns of geometric objects. In: Horizons of Combinatorics, Bolyai Soc. Math. Stud., vol. 17, pp. 79–103. Springer, Berlin (2008)CrossRefGoogle Scholar
  15. 15.
    Ghosh, S.K.: Visibility algorithms in the plane. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  16. 16.
    Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in IRd. Discrete Comput. Geom. 1(3), 219–227 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goodman, J.E., Pollack, R.: Allowable sequences and order types in discrete and computational geometry. In: New Trends in Discrete and Computational Geometry, Algorithms Combin., vol. 10, pp. 103–134. Springer, Berlin (1993)CrossRefGoogle Scholar
  18. 18.
    Matoušek, J.: Blocking visibility for points in general position. Discrete & Computational Geometry 42(2), 219–223 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Matoušek, J., Valtr, P.: The complexity of lower envelope of segments with h endpoints. Intuitive Geometry, Bolyai Society of Math. Studies 6, 407–411 (1997)MathSciNetzbMATHGoogle Scholar
  20. 20.
    O’Rourke, J.: Visibility. In: Handbook of Discrete and Computational Geometry. CRC Press Ser. Discrete Math. Appl, pp. 467–479. CRC, Boca Raton (1997)Google Scholar
  21. 21.
    O’Rourke, J.: Open problems in the combinatorics of visibility and illumination. In: Advances in Discrete and Computational Geometry, South Hadley, MA. Contemp. Math., vol. 223, pp. 237–243. Amer. Math. Soc., Providence (1999)CrossRefGoogle Scholar
  22. 22.
    Pach, J.: Midpoints of segments induced by a point set. Geombinatorics 13(2), 98–105 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York (1995)CrossRefzbMATHGoogle Scholar
  24. 24.
    Prömel, H.J., Steger, A.: Excluding induced subgraphs: Quadrilaterals. Random Structures and Algorithms 2(1), 55–71 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Prömel, H.J., Steger, A.: Excluding induced subgraphs III: A general asymptotic. Random Structures and Algorithms 3(1), 19–31 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Prömel, H.J., Steger, A.: Excluding induced subgraphs II: extremal graphs. Discrete Applied Mathematics 44, 283–294 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Szemerédi, E., Trotter Jr., W.T.: A combinatorial distinction between the Euclidean and projective planes. European J. Combin. 4(4), 385–394 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Szemerédi, E., Trotter Jr., W.T.: Extremal problems in discrete geometry. Combinatorica 3(3-4), 381–392 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tucker, A.: Coloring a family of circular arcs. SIAM Journal on Applied Mathematics 29(3), 493–502 (1975), http://www.jstor.org/stable/2100446 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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) (in Russian)MathSciNetGoogle Scholar
  31. 31.
    Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, pp. 973–1027. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.RutgersThe State University of New JerseyPiscatawayUSA
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.The Graduate Center of the City University of New YorkNew YorkUSA

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