# 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.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Alon, N.: The number of polytopes, configurations and real matroids. Mathematika 33(1), 62–71 (1986)
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)
5. 5.
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. Algorithms and Applications, 2nd edn. Springer, Berlin (2000)
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)
8. 8.
Erdős, P., Hajnal, A.: Ramsey-type theorems. Discrete Appl. Math. 25(1-2), 37–52 (1989)
9. 9.
Erdős, P.: Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53, 292–294 (1947)
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)
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)
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)
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)
15. 15.
Ghosh, S.K.: Visibility algorithms in the plane. Cambridge University Press, Cambridge (2007)
16. 16.
Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in IRd. Discrete Comput. Geom. 1(3), 219–227 (1986)
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)
18. 18.
Matoušek, J.: Blocking visibility for points in general position. Discrete & Computational Geometry 42(2), 219–223 (2009)
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)
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)
22. 22.
Pach, J.: Midpoints of segments induced by a point set. Geombinatorics 13(2), 98–105 (2003)
23. 23.
Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York (1995)
24. 24.
Prömel, H.J., Steger, A.: Excluding induced subgraphs: Quadrilaterals. Random Structures and Algorithms 2(1), 55–71 (1991)
25. 25.
Prömel, H.J., Steger, A.: Excluding induced subgraphs III: A general asymptotic. Random Structures and Algorithms 3(1), 19–31 (1992)
26. 26.
Prömel, H.J., Steger, A.: Excluding induced subgraphs II: extremal graphs. Discrete Applied Mathematics 44, 283–294 (1993)
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)
28. 28.
Szemerédi, E., Trotter Jr., W.T.: Extremal problems in discrete geometry. Combinatorica 3(3-4), 381–392 (1983)
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
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)
31. 31.
Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, pp. 973–1027. North-Holland, Amsterdam (2000)