Drawing Graphs Using a Small Number of Obstacles

  • Martin BalkoEmail author
  • Josef Cibulka
  • Pavel Valtr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number \({{\mathrm{obs}}}(G)\) of G is the minimum number of obstacles in an obstacle representation of G.

We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph G satisfies \({{\mathrm{obs}}}(G) \le 2n\log {n}\). This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For bipartite n-vertex graphs, we improve this bound to \(n-1\). Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound \(2^{\varOmega (hn)}\) on the number of n-vertex graphs with obstacle number at most h for \(h<n\) and an asymptotically matching lower bound \(\varOmega (n^{4/3}M^{2/3})\) for the complexity of a collection of \(M \ge \varOmega (n)\) faces in an arrangement of \(n^2\) line segments with 2n endpoints.


Obstacle number Geometric drawing Obstacle representation Arrangement of line segments 


  1. 1.
    Alpert, H., Koch, C., Laison, J.D.: Obstacle numbers of graphs. Discrete Comput. Geom. 44(1), 223–244 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    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(1), 257–270 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aronov, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: The number of edges of many faces in a line segment arrangement. Combinatorica 12(3), 261–274 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Dujmović, V., Morin, P.: On obstacle numbers. Electron. J. Combin. 22(3), P3.1 (2015)Google Scholar
  5. 5.
    Edelsbrunner, H., Welzl, E.: On the maximal number of edges of many faces in an arrangement. J. Combin. Theory Ser. A 41(2), 159–166 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fulek, R., Saeedi, N., Sarıöz, D.: Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 249–261. Springer, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 5th edn. Clarendon Press, Oxford (1979)zbMATHGoogle Scholar
  8. 8.
    Matoušek, J., Valtr, P.: The complexity of lower envelope of segments with \(h\) endpoints. Intuitive Geom. Bolyai Soc. Math. Stud. 6, 407–411 (1997)Google Scholar
  9. 9.
    Mukkamala, P., Pach, J., Pálvölgyi, D.: Lower bounds on the obstacle number of graphs. Electron. J. Combin. 19(2), P32 (2012)Google Scholar
  10. 10.
    Mukkamala, P., Pach, J., Sarıöz, D.: Graphs with large obstacle numbers. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 292–303. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  11. 11.
    Pach, J., Sarıöz, D.: On the structure of graphs with low obstacle number. Graphs Combin. 27(3), 465–473 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Valtr, P.: Convex independent sets and 7-holes in restricted planar point sets. Discrete Comput. Geom. 7(1), 135–152 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Wiernik, A., Sharir, M.: Planar realizations of nonlinear Davenport-Schinzel sequences by segments. Discrete Comput. Geom. 3(1), 15–47 (1988)zbMATHMathSciNetCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic

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