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# Recognition and Complexity of Point Visibility Graphs

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LIPIcs.SOCG.2015.171.pdf
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## Cite As

Jean Cardinal and Udo Hoffmann. Recognition and Complexity of Point Visibility Graphs. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 171-185, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.SOCG.2015.171

## Abstract

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 Mnev 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 having integer coordinates.
##### Keywords
• point visibility graphs
• recognition
• existential theory of the reals

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## References

1. James Abello and Krishna Kumar. Visibility graphs and oriented matroids. Discrete & Computational Geometry, 28(4):449-465, 2002.
2. Karim A. Adiprasito, Arnau Padrol, and Louis Theran. Universality theorems for inscribed polytopes and Delaunay triangulations. ArXiv e-prints, 2014.
3. Daniel Bienstock. Some provably hard crossing number problems. Discrete and Computational Geometry, 6:443-459, 1991.
4. John Canny. Some algebraic and geometric computations in PSPACE. In STOC '88, pages 460-467. ACM, 1988.
5. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2008 (third edition).
6. Subir K. Ghosh. On recognizing and characterizing visibility graphs of simple polygons. In 1st Scandinavian Workshop on Algorithm Theory (SWAT), pages 96-104, 1988.
7. Subir K. Ghosh. On recognizing and characterizing visibility graphs of simple polygons. Discrete & Computational Geometry, 17(2):143-162, 1997.
8. Subir K. Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007.
9. Subir K. Ghosh and Partha P. Goswami. Unsolved problems in visibility graphs of points, segments, and polygons. ACM Computing Surveys (CSUR), 46(2):22, 2013.
10. Subir K. Ghosh and Bodhayan Roy. Some results on point visibility graphs. In Algorithms and Computation (WALCOM), volume 8344 of Lecture Notes in Computer Science, pages 163-175. Springer, 2014.
11. Jacob E. Goodman, Richard Pollack, and Bernd Sturmfels. The intrinsic spread of a configuration in ℝ^d. Journal of the American Mathematical Society, pages 639-651, 1990.
12. Branko Grünbaum. Convex Polytopes, volume 221 (2nd ed.) of Graduate Texts in Mathematics. Springer-Verlag, 2003.
13. Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41:1051-1107, 2002.
14. Jan Kára, Attila Pór, and David R. Wood. On the chromatic number of the visibility graph of a set of points in the plane. Discrete & Computational Geometry, 34(3):497-506, 2005.
15. Jan Kratochvíl and Jirí Matoušek. Intersection graphs of segments. Journal of Combinatorial Theory. Series B, 62(2):289-315, 1994.
16. Jan Kynčl. Simple realizability of complete abstract topological graphs in P. Discrete and Computational Geometry, 45(3):383-399, 2011.
17. Tomás Lozano-Pérez and Michael A. Wesley. An algorithm for planning collision-free paths among polyhedral obstacles. Commun. ACM, 22(10):560-570, October 1979.
18. Jiří Matoušek. Intersection graphs of segments and ∃ℝ. ArXiv e-prints, 2014.
19. Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. Journal of Combinatorial Theory, Series B, 103(1):114 - 143, 2013.
20. Nicolai E. Mnëv. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and geometry—Rohlin seminar, pages 527-543. Springer, 1988.
21. Joseph O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987.
22. Joseph O'Rourke and Ileana Streinu. Vertex-edge pseudo-visibility graphs: Characterization and recognition. In Symposium on Computational Geometry, pages 119-128, 1997.
23. Michael S. Payne, Attila Pór, Pavel Valtr, and David R. Wood. On the connectivity of visibility graphs. Discrete & Computational Geometry, 48(3):669-681, 2012.
24. Attila Pór and David R. Wood. On visibility and blockers. JoCG, 1(1):29-40, 2010.
25. Jürgen Richter-Gebert. Mnëv’s universality theorem revisited. In Proceedings of the Séminaire Lotharingien de Combinatoire, pages 211-225, 1995.
26. Bodhayan Roy. Point visibility graph recognition is NP-hard. ArXiv e-prints, 2014.
27. Marcus Schaefer. Complexity of some geometric and topological problems. In 17th International Symposium on Graph Drawing (GD), pages 334-344, 2009.
28. Marcus Schaefer. Realizability of graphs and linkages. In Thirty Essays on Geometric Graph Theory. Springer, 2012.
29. Peter W. Shor. Stretchability of pseudolines is NP-hard. Applied Geometry and Discrete Mathematics-The Victor Klee Festschrift, 4:531-554, 1991.
30. Karl Georg Christian Staudt. Geometrie der Lage. Verlag von Bauer und Raspe, 1847.
31. Ileana Streinu. Non-stretchable pseudo-visibility graphs. Comput. Geom., 31(3):195-206, 2005.
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