Discrete & Computational Geometry

, Volume 57, Issue 1, pp 164–178 | Cite as

Recognition and Complexity of Point Visibility Graphs

  • Jean Cardinal
  • Udo Hoffmann


A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: Given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnëv and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points on a grid. We also prove that the problem of recognizing visibility graphs of points on a grid is decidable if and only if the existential theory of the rationals is decidable.


Point visibility graphs Existential theory of the reals Geometric graph representations 



We thank an anonymous SoCG referee for pointing out an error in the original proof.We also thank the anonymous DCG referee for his numerous suggestions, which helped to simplify the constructions significantly compared to the conference version. J. Cardinal is supported by the ARC (Action de Recherche Concertée) project COPHYMA. U. Hoffmann is supported by the Deutsche Forschungsgemeinschaft within the research training group ’Methods for Discrete Structures’ (GRK 1408).


  1. 1.
    Abello, J., Kumar, K.: Visibility graphs and oriented matroids. Discrete Comput. Geom. 28, 449–465 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adiprasito, K.A., Padrol, A., Theran, L.: Universality theorems for inscribed polytopes and Delaunay triangulations. Discrete Comput. Geom. 54, 412–431 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bienstock, D.: Some provably hard crossing number problems. Discrete Comput. Geom. 6, 443–459 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids, 2nd edn. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar
  5. 5.
    Canny, J.: Some algebraic and geometric computations in PSPACE. In: STOC, pp. 460–467. ACM, New York (1988)Google Scholar
  6. 6.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  7. 7.
    Ghosh, S.K.: On recognizing and characterizing visibility graphs of simple polygons. Discrete Comput. Geom. 17, 143–162 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Ghosh, S.K., Goswami, P.P.: Unsolved problems in visibility graphs of points, segments, and polygons. ACM Comput. Surv. 46, 22 (2013)CrossRefMATHGoogle Scholar
  10. 10.
    Ghosh, S.K., Roy, B.: Some results on point visibility graphs. In: WALCOM. LNCS, vol. 8344, pp. 163–175. Springer, Berlin (2014)Google Scholar
  11. 11.
    Goodman, J.E., Pollack, R., Sturmfels, B.: The intrinsic spread of a configuration in \(\mathbb{R}^d\). J. Am. Math. Soc. 3, 639–651 (1990)MathSciNetMATHGoogle Scholar
  12. 12.
    Grünbaum, B.: Arrangements and Spreads. Regional Conference Series in Mathematics, vol. 10. AMS, Providence, RI (1972)Google Scholar
  13. 13.
    Grünbaum, B.: Convex Polytopes. Graduate Texts in Mathematics, vol. 221, 2nd edn. Springer, New York (2003)Google Scholar
  14. 14.
    Kapovich, M., Millson, J.J.: Universality theorems for configuration spaces of planar linkages. Topology 41, 1051–1107 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kára, J., Pór, A., Wood, D.R.: On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom. 34, 497–506 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Comb. Theory, Ser. B 62, 289–315 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kynčl, J.: Simple realizability of complete abstract topological graphs in P. Discrete Comput. Geom. 45, 383–399 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lozano-Pérez, T., Wesley, M.A.: An algorithm for planning collision-free paths among polyhedral obstacles. Commun. ACM 22, 560–570 (1979)CrossRefGoogle Scholar
  19. 19.
    Matiyasevich, Y.V.: Enumerable sets are diophantine. Dokl. Akad. Nauk SSSR 191, 279–282 (1970)MathSciNetMATHGoogle Scholar
  20. 20.
    McDiarmid, C., Müller, T.: Integer realizations of disk and segment graphs. J. Comb. Theory, Ser. B 103, 114–143 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and Geometry—Rohlin Seminar. LNM, pp. 527–543. Springer, Berlin (1988)Google Scholar
  22. 22.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, New York (1987)MATHGoogle Scholar
  23. 23.
    O’Rourke, J., Streinu, I.: Vertex-edge pseudo-visibility graphs: characterization and recognition. In: SoCG, pp. 119–128. ACM, New York (1997)Google Scholar
  24. 24.
    Payne, M.S., Pór, A., Valtr, P., Wood, D.R.: On the connectivity of visibility graphs. Discrete Comput. Geom. 48, 669–681 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pór, A., Wood, D.R.: On visibility and blockers. J. Comput. Geom. 1, 29–40 (2010)MathSciNetGoogle Scholar
  26. 26.
    Roy, B.: Point visibility graph recognition is NP-hard. arXiv:1406.2428 (2014)
  27. 27.
    Schaefer, M.: Complexity of some geometric and topological problems. In: GD, LNCS, vol. 5849, pp. 334–344. Springer, Berlin (2009)Google Scholar
  28. 28.
    Schaefer, M.: Realizability of graphs and linkages. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory. Springer, Berlin (2012)Google Scholar
  29. 29.
    Shor, P.W.: Stretchability of pseudolines is NP-hard. Appl. Geom. Discrete Math. 4, 531–554 (1991)MathSciNetMATHGoogle Scholar
  30. 30.
    Staudt, K.G.C.: Geometrie der Lage. F. Korn, Nuremberg (1847)Google Scholar
  31. 31.
    Streinu, I.: Non-stretchable pseudo-visibility graphs. Comput. Geom. 31, 195–206 (2005)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Sturmfels, B.: On the decidability of diophantine problems in combinatorial geometry. Bull. Am. Math. Soc. 17, 121–124 (1987)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Université libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.TU BerlinBerlinGermany

Personalised recommendations