Document Open Access Logo

Recognizing Penny and Marble Graphs Is Hard for Existential Theory of the Reals

Authors Anna Lubiw , Marcus Schaefer

PDF

File

Thumbnail PDF
PDF
LIPIcs.GD.2026.8.pdf
  • Filesize: 28.61 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • penny graphs
  • contact graphs of unit disks
  • marble graphs
  • contact graphs of balls
  • ∃ℝ
  • existential theory of the reals
  • computational complexity
  • rigidity

Metrics

Abstract

We show that the recognition problem for penny graphs (contact graphs of unit disks in the plane) is ∃ℝ-complete, that is, computationally as hard as the existential theory of the reals, even if a combinatorial plane embedding of the graph is given. The exact complexity of the penny graph recognition problem has been a long-standing open problem. 
We lift the penny graph result to three dimensions and show that the recognition problem for marble graphs (contact graphs of unit balls in three dimensions) is ∃ℝ-complete. 
Finally, we show that rigidity of penny graphs is ∀ℝ-complete and look at grid embeddings of penny graphs that are trees.

Cite As Get BibTex

Anna Lubiw and Marcus Schaefer. Recognizing Penny and Marble Graphs Is Hard for Existential Theory of the Reals. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.8

Author Details

Anna Lubiw
  • University of Waterloo, Canada
Marcus Schaefer
  • DePaul University, Chicago, IL, USA

Acknowledgements

We would like to thank the anonymous referees for useful feedback.

References

  1. Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Jayson Lynch, and Tao B. Schardl. Who needs crossings? Noncrossing linkages are universal, and deciding (global) rigidity is hard. Journal of Computational Geometry, 16(1):378-452, June 2025. URL: https://doi.org/10.20382/jocg.v16i1a12.
  2. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. The art gallery problem is ∃ℝ-complete. J. ACM, 69(1):Art. 4, 70, 2022. URL: https://doi.org/10.1145/3486220.
  3. Mikkel Abrahamsen and Tillmann Miltzow. Dynamic toolbox for ETRINV. arXiv, abs/1912.08674, 2019. URL: https://arxiv.org/abs/1912.08674.
  4. Daniel Bienstock. Some provably hard crossing number problems. Discrete Comput. Geom., 6(5):443-459, 1991. URL: https://doi.org/10.1007/BF02574701.
  5. Clinton Bowen, Stephane Durocher, Maarten Löffler, Anika Rounds, André Schulz, and Csaba D. Tóth. Realization of simply connected polygonal linkages and recognition of unit disk contact trees. In Graph Drawing and Network Visualization, volume 9411 of Lecture Notes in Comput. Sci., pages 447-459. Springer, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-27261-0_37.
  6. Heinz Breu and David G. Kirkpatrick. On the complexity of recognizing intersection and touching graphs of disks. In Graph Drawing (Passau, 1995), volume 1027 of Lecture Notes in Comput. Sci., pages 88-98. Springer, Berlin, 1996. URL: https://doi.org/10.1007/BFb0021793.
  7. Heinz Breu and David G. Kirkpatrick. Unit disk graph recognition is NP-hard. Comput. Geom., 9(1-2):3-24, 1998. URL: https://doi.org/10.1016/S0925-7721(97)00014-X.
  8. John Bryant and Chris Sangwin. How round is Your Circle?: Where Engineering and Mathematics Meet. Princeton University Press, 2011. Google Scholar
  9. Sergio Cabello, Erik D. Demaine, and Günter Rote. Planar embeddings of graphs with specified edge lengths. J. Graph Algorithms Appl., 11(1):259-276, 2007. URL: https://doi.org/10.7155/jgaa.00145.
  10. John Canny. Some algebraic and geometric computations in pspace. In STOC '88: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pages 460-469, New York, NY, USA, 1988. ACM. URL: https://doi.org/10.1145/62212.62257.
  11. Man-Kwun Chiu, Jonas Cleve, and Martin Nöllenburg. Recognizing embedded caterpillars with weak unit disk contact representations is NP-hard. arXiv, 2020. URL: https://arxiv.org/abs/2010.01881.
  12. Jonas Cleve. Weak unit disk contact representations for graphs without embedding. arXiv, 2020. URL: https://arxiv.org/abs/2010.01886.
  13. Peter Eades and Sue Whitesides. The logic engine and the realization problem for nearest neighbor graphs. Theoret. Comput. Sci., 169(1):23-37, 1996. URL: https://doi.org/10.1016/S0304-3975(97)84223-5.
  14. Peter Eades and Nicholas C. Wormald. Fixed edge-length graph drawing is NP-hard. Discrete Appl. Math., 28(2):111-134, 1990. URL: https://doi.org/10.1016/0166-218X(90)90110-X.
  15. P. Erdös. On sets of distances of n points. Amer. Math. Monthly, 53:248-250, 1946. URL: https://doi.org/10.2307/2305092.
  16. Richard K. Guy and Robert E. Woodrow, editors. The Lighter Side of Mathematics, MAA Spectrum. Mathematical Association of America, Washington, DC, 1994. Google Scholar
  17. Heiko Harborth. Lösung zu Problem 664a. Elem. Math., 29:14-15, 1974. Google Scholar
  18. Petr Hliněný. Touching graphs of unit balls. In Graph Drawing (Rome, 1997), volume 1353 of Lecture Notes in Comput. Sci., pages 350-358. Springer, Berlin, 1997. URL: https://doi.org/10.1007/3-540-63938-1_80.
  19. Petr Hliněný and Jan Kratochvíl. Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Math., 229(1-3):101-124, 2001. URL: https://doi.org/10.1016/S0012-365X(00)00204-1.
  20. Paul Koebe. Kontaktprobleme der konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl., 88:141-164, 1936. Google Scholar
  21. Jan Kratochvíl and Jiří Matoušek. Intersection graphs of segments. J. Combin. Theory Ser. B, 62(2):289-315, 1994. URL: https://doi.org/10.1006/jctb.1994.1071.
  22. Jérémy Lavollée and Konrad Swanepoel. A tight bound for the number of edges of matchstick graphs. Discrete Comput. Geom., 72(4):1530-1544, 2024. URL: https://doi.org/10.1007/s00454-023-00530-z.
  23. Cédric Lorand. K5 and K3,3 are toroidal penny graphs. arXiv, 2024. URL: https://arxiv.org/abs/2410.10673.
  24. Anna Lubiw and Marcus Schaefer. Recognizing penny and marble graphs is hard for existential theory of the reals. arXiv, 2025. URL: https://doi.org/10.48550/arXiv.2508.10136.
  25. Hiroshi Maehara. Distances in a rigid unit-distance graph in the plane. Discrete Appl. Math., 31(2):193-200, 1991. URL: https://doi.org/10.1016/0166-218X(91)90070-D.
  26. Jiří Matoušek. Intersection graphs of segments and ∃ℝ. In János Pach and Géza Tóth, editors, Courses in Discrete and Computational Geometry, pages 129-165. Springer Nature Switzerland, Cham, 2026. URL: https://doi.org/10.1007/978-3-032-10503-5_4.
  27. Colin McDiarmid and Tobias Müller. The number of bits needed to represent a unit disk graph. In Graph-Theoretic Concepts in Computer Science, volume 6410 of Lecture Notes in Comput. Sci., pages 315-323. Springer, Berlin, 2010. URL: https://doi.org/10.1007/978-3-642-16926-7_29.
  28. Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. J. Combin. Theory Ser. B, 103(1):114-143, 2013. URL: https://doi.org/10.1016/j.jctb.2012.09.004.
  29. Bojan Mohar and Carsten Thomassen. Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 2001. Google Scholar
  30. Marcus Schaefer. Realizability of graphs and linkages. In Thirty Essays on Geometric Graph Theory, pages 461-482. Springer, New York, 2013. URL: https://doi.org/10.1007/978-1-4614-0110-0_24.
  31. Marcus Schaefer, Jean Cardinal, and Tillmann Miltzow. The existential theory of the reals as a complexity class: A compendium. In János Pach and Géza Tóth, editors, Courses in Discrete and Computational Geometry, pages 167-313. Springer Nature Switzerland, Cham, 2026. URL: https://doi.org/10.1007/978-3-032-10503-5_5.
  32. Peter W. Shor. Stretchability of pseudolines is NP-hard. In Applied Geometry and Discrete Mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 531-554. Amer. Math. Soc., Providence, RI, 1991. URL: https://doi.org/10.1090/dimacs/004/41.
  33. Kenneth Stephenson. Circle packing: a mathematical tale. Notices Amer. Math. Soc., 50(11):1376-1388, 2003. Google Scholar
  34. Sue Whitesides. Algorithmic issues in the geometry of planar linkage movement. Aust. Comput. J., 24(2):42-50, 1992. Google Scholar
  35. Yechiam Yemini. Some theoretical aspects of position-location problems. In 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979), pages 1-8. IEEE, New York, 1979. URL: https://doi.org/10.1109/SFCS.1979.39.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact