Document Open Access Logo

Compact Representation of Semilinear and Terrain-Like Graphs

Authors Jean Cardinal , Yelena Yuditsky

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.67.pdf
  • Filesize: 0.85 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Graph algorithms analysis
Keywords
  • Biclique covers
  • intersection graphs
  • visibility graphs
  • Zarankiewicz’s problem

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We consider the existence and construction of biclique covers of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The size of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size.
In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size O(npolylog n). This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz’s problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (Forum Math. Sigma, 2021).
We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size O(nlog³ n). This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (Discrete Comput. Geom. 1994).
Finally, we prove that there exists families of unit disk graphs on n vertices that do not admit biclique coverings of size o(n^{4/3}), showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.

Cite As Get BibTex

Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.67

Author Details

Jean Cardinal
  • Université libre de Bruxelles (ULB), Belgium
Yelena Yuditsky
  • Université libre de Bruxelles (ULB), Belgium

Funding

  • Cardinal, Jean: Jean Cardinal was supported by the Fonds de la Recherche Scientifique - FNRS under Grant n° T003325F.
  • Yuditsky, Yelena: Yelena Yuditsky was supported by the Fonds de la Recherche Scientifique - FNRS as a Postdoctoral Researcher.

Acknowledgements

This work was inspired by discussions at the Tenth Annual Workshop on Geometry and Graphs (WoGaG’23), held at the Bellairs Research Institute, (McGill), Barbados, in February 2023.

References

  1. James Abello, Ömer Eğecioğlu, and Krishna Kumar. Visibility graphs of staircase polygons and the weak Bruhat order. I. From visibility graphs to maximal chains. Discrete Comput. Geom., 14(3):331-358, 1995. URL: https://doi.org/10.1007/BF02570710.
  2. Eyal Ackerman and Balázs Keszegh. The Zarankiewicz Problem for Polygon Visibility Graphs. arXiv preprint arXiv:2503.09115, 2025. URL: https://doi.org/10.48550/arXiv.2503.09115.
  3. Pankaj K. Agarwal, Noga Alon, Boris Aronov, and Subash Suri. Can visibility graphs be represented compactly? Discrete Comput. Geom., 12(3):347-365, 1994. URL: https://doi.org/10.1007/BF02574385.
  4. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection queries for flat semi-algebraic objects in three dimensions and related problems. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 4:1-4:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.SOCG.2022.4.
  5. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection queries for flat semi-algebraic objects in three dimensions and related problems. arXiv preprint 2203.10241, 2025. URL: https://doi.org/10.48550/arXiv.2203.10241.
  6. Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. Contemporary Mathematics, 223(1):56, 1999. URL: https://doi.org/10.1090/conm/223/03131.
  7. Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Semi-algebraic off-line range searching and biclique partitions in the plane. In Wolfgang Mulzer and Jeff M. Phillips, editors, 40th International Symposium on Computational Geometry, SoCG 2024, June 11-14, 2024, Athens, Greece, volume 293 of LIPIcs, pages 4:1-4:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.SOCG.2024.4.
  8. Pankaj K. Agarwal and Kasturi R. Varadarajan. Efficient algorithms for approximating polygonal chains. Discrete Comput. Geom., 23(2):273-291, 2000. URL: https://doi.org/10.1007/PL00009500.
  9. Adeli A. Ahmadlou M., Adeli H. New diagnostic EEG markers of the Alzheimer’s disease using visibility graph. J. Neural Transm., 117:1099-1109, 2010. URL: https://doi.org/10.1007/s00702-010-0450-3.
  10. Noga Alon, János Pach, Rom Pinchasi, Radoš Radoičić, and Micha Sharir. Crossing patterns of semi-algebraic sets. J. Combin. Theory Ser. A, 111(2):310-326, 2005. URL: https://doi.org/10.1016/j.jcta.2004.12.008.
  11. Safwa Ameer, Matt Gibson-Lopez, Erik Krohn, Sean Soderman, and Qing Wang. Terrain visibility graphs: persistence is not enough. In 36th International Symposium on Computational Geometry, volume 164 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 6, 13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.SOCG.2020.6.
  12. Stav Ashur, Omrit Filtser, Matthew J. Katz, and Rachel Saban. Terrain-like graphs: PTASs for guarding weakly-visible polygons and terrains. Comput. Geom., 101:Paper No. 101832, 13, 2022. URL: https://doi.org/10.1016/j.comgeo.2021.101832.
  13. Abdul Basit, Artem Chernikov, Sergei Starchenko, Terence Tao, and Chieu-Minh Tran. Zarankiewicz’s problem for semilinear hypergraphs. In Forum of Mathematics, Sigma, volume 9, page e59. Cambridge University Press, 2021. URL: https://doi.org/10.1017/fms.2021.52.
  14. Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth. Sparse bounded hop-spanners for geometric intersection graphs. arXiv preprint arXiv:2504.05861, 2025. URL: https://doi.org/10.48550/arXiv.2504.05861.
  15. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable fo model checking. ACM Journal of the ACM (JACM), 69(1):1-46, 2021. URL: https://doi.org/10.1145/3486655.
  16. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. URL: https://doi.org/10.1137/1.9780898719796.
  17. Peter Braß and Christian Knauer. On counting point-hyperplane incidences. Computational Geometry, 25(1-2):13-20, 2003. URL: https://doi.org/10.1016/S0925-7721(02)00127-X.
  18. Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, and Christian Knauer. Geometric matching and bottleneck problems. In 40th International Symposium on Computational Geometry, SoCG 2024, June 11-14, 2024, Athens, Greece, pages 31:1-31:15, 2024. URL: https://doi.org/10.4230/LIPICS.SOCG.2024.31.
  19. Jean Cardinal and Micha Sharir. Improved algebraic degeneracy testing. Discrete & Computational Geometry, pages 1-19, 2024. URL: https://doi.org/10.1007/s00454-024-00673-7.
  20. Jean Cardinal and Yelena Yuditsky. Compact representation of semilinear and terrain-like graphs. arXiv preprint arXiv:2507.00252, 2025. URL: https://doi.org/10.48550/arXiv.2507.00252.
  21. Timothy M. Chan. Dynamic subgraph connectivity with geometric applications. SIAM Journal on Computing, 36(3):681-694, 2006. URL: https://doi.org/10.1137/S009753970343912X.
  22. Timothy M. Chan. All-pairs shortest paths with real weights in O(n³/log n) Time. Algorithmica, 50(2):236-243, 2008. URL: https://doi.org/10.1007/S00453-007-9062-1.
  23. Timothy M. Chan. Optimal partition trees. Discrete Comput. Geom., 47(4):661-690, 2012. URL: https://doi.org/10.1007/s00454-012-9410-z.
  24. Timothy M Chan, Pingan Cheng, and Da Wei Zheng. Semialgebraic range stabbing, ray shooting, and intersection counting in the plane. In 40th International Symposium on Computational Geometry, SoCG 2024, page 33. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2024. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.33.
  25. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in geometric intersection graphs. J. Comput. Geom., 10(1):27-41, 2019. URL: https://doi.org/10.20382/JOCG.V10I1A2.
  26. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom., 9(2):145-158, 1993. URL: https://doi.org/10.1007/BF02189314.
  27. Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, and Micha Sharir. Algorithms for bichromatic line-segment problems polyhedral terrains. Algorithmica, 11(2):116-132, 1994. URL: https://doi.org/10.1007/BF01182771.
  28. Bernard Chazelle and Leonidas J. Guibas. Visibility and intersection problems in plane geometry. Discret. Comput. Geom., 4:551-581, 1989. URL: https://doi.org/10.1007/BF02187747.
  29. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 612-623. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00064.
  30. Fan R. K. Chung, Paul Erdős, and Joel Spencer. On the decomposition of graphs into complete bipartite subgraphs. In Studies in pure mathematics, pages 95-101. Birkhäuser, Basel, 1983. URL: https://doi.org/10.1007/978-3-0348-5438-2_10.
  31. David Conlon. Some remarks on the Zarankiewicz problem. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 173(1), pages 155-161. Cambridge University Press, 2022. URL: https://doi.org/10.1017/S0305004121000475.
  32. David Conlon, Jacob Fox, János Pach, Benny Sudakov, and Andrew Suk. Ramsey-type results for semi-algebraic relations. Trans. Amer. Math. Soc., 366(9):5043-5065, 2014. URL: https://doi.org/10.1090/S0002-9947-2014-06179-5.
  33. Jonathan B. Conroy and Csaba D. Tóth. Hop-spanners for geometric intersection graphs. J. Comput. Geom., 14(2):26-64, 2022. URL: https://doi.org/10.20382/JOCG.V14I2A3.
  34. James Davies, Tomasz Krawczyk, Rose McCarty, and Bartosz Walczak. Coloring polygon visibility graphs and their generalizations. J. Combin. Theory Ser. B, 161:268-300, 2023. URL: https://doi.org/10.1016/j.jctb.2023.02.010.
  35. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  36. Sarita de Berg, Nathan van Beusekom, Max van Mulken, Kevin Verbeek, and Jules Wulms. Competitive searching over terrains. In José A. Soto and Andreas Wiese, editors, LATIN 2024: Theoretical Informatics - 16th Latin American Symposium, Puerto Varas, Chile, March 18-22, 2024, Proceedings, Part I, volume 14578 of Lecture Notes in Computer Science, pages 254-269. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-55598-5_17.
  37. Thao Do. Representation complexities of semialgebraic graphs. SIAM J. Discrete Math., 33(4):1864-1877, 2019. URL: https://doi.org/10.1137/18M1221606.
  38. Herbert Edelsbrunner. Algorithms in combinatorial geometry, volume 10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1987. URL: https://doi.org/10.1007/978-3-642-61568-9.
  39. Jeff Erickson. New lower bounds for Hopcroft’s problem. Discrete & Computational Geometry, 16(4):389-418, 1996. URL: https://doi.org/10.1007/BF02712875.
  40. William Evans and Noushin Saeedi. On characterizing terrain visibility graphs. J. Comput. Geom., 6(1):108-141, 2015. URL: https://doi.org/10.20382/jocg.v6i1a5.
  41. Tomás Feder and Rajeev Motwani. Clique partitions, graph compression and speeding-up algorithms. J. Comput. System Sci., 51(2):261-272, 1995. URL: https://doi.org/10.1006/jcss.1995.1065.
  42. Jacob Fox, János Pach, Adam Sheffer, Andrew Suk, and Joshua Zahl. A semi-algebraic version of Zarankiewicz’s problem. J. Eur. Math. Soc. (JEMS), 19(6):1785-1810, 2017. URL: https://doi.org/10.4171/JEMS/705.
  43. Vincent Froese and Malte Renken. A fast shortest path algorithm on terrain-like graphs. Discrete Comput. Geom., 66(2):737-750, 2021. URL: https://doi.org/10.1007/s00454-020-00226-8.
  44. Vincent Froese and Malte Renken. Persistent graphs and cyclic polytope triangulations. Comb., 41(3):407-423, 2021. URL: https://doi.org/10.1007/S00493-020-4369-5.
  45. Vincent Froese and Malte Renken. Terrain-like graphs and the median Genocchi numbers. European J. Combin., 115:Paper No. 103780, 8, 2024. URL: https://doi.org/10.1016/j.ejc.2023.103780.
  46. Subir K. Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511543340.
  47. Oktay Günlük. A new min-cut max-flow ratio for multicommodity flows. SIAM Journal on Discrete Mathematics, 21(1):1-15, 2007. URL: https://doi.org/10.1137/S089548010138917X.
  48. Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Ann. of Math. (2), 181(1):155-190, 2015. URL: https://doi.org/10.4007/annals.2015.181.1.2.
  49. Pavol Hell, Jing Huang, Jephian C.-H. Lin, and Ross M. McConnell. Bipartite analogues of comparability and cocomparability graphs. SIAM J. Discrete Math., 34(3):1969-1983, 2020. URL: https://doi.org/10.1137/19M1263789.
  50. Jing Huang. Representation characterizations of chordal bipartite graphs. J. Combin. Theory Ser. B, 96(5):673-683, 2006. URL: https://doi.org/10.1016/j.jctb.2006.01.001.
  51. Stasys Jukna and Alexander S Kulikov. On covering graphs by complete bipartite subgraphs. Discrete Mathematics, 309(10):3399-3403, 2009. URL: https://doi.org/10.1016/J.DISC.2008.09.036.
  52. Prahlad Narasimhan Kasthurirangan. One-sided terrain guarding and chordal graphs. Discrete Appl. Math., 348:192-201, 2024. URL: https://doi.org/10.1016/j.dam.2024.01.034.
  53. Matthew J. Katz, Rachel Saban, and Micha Sharir. Near-linear algorithms for visibility graphs over a 1.5-dimensional terrain. In Timothy Chan, Johannes Fischer, John Iacono, and Grzegorz Herman, editors, 32nd Annual European Symposium on Algorithms (ESA 2024), volume 308 of Leibniz International Proceedings in Informatics (LIPIcs), pages 77:1-77:17, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.77.
  54. Matthew J. Katz and Micha Sharir. An expander-based approach to geometric optimization. SIAM J. Comput., 26(5):1384-1408, 1997. URL: https://doi.org/10.1137/S0097539794268649.
  55. Oliver Kullmann and Ankit Shukla. Transforming quantified boolean formulas using biclique covers. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pages 372-390. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-30820-8_23.
  56. Lucas Lacasa, Bartolo Luque, Fernando Ballesteros, Jordi Luque, and Juan Carlos Nuño. From time series to complex networks: The visibility graph. Proceedings of the National Academy of Sciences, 105(13):4972-4975, 2008. URL: https://doi.org/10.1073/pnas.0709247105.
  57. Jiří Matoušek. Range searching with efficient hierarchical cuttings. In Proceedings of the eighth annual symposium on Computational geometry, pages 276-285, 1992. URL: https://doi.org/10.1145/142675.142732.
  58. Jiří Matoušek. Cutting hyperplane arrangements. Discrete Comput. Geom., 6(5):385-406, 1991. URL: https://doi.org/10.1007/BF02574697.
  59. Jiří Matoušek. Efficient partition trees. Discrete Comput. Geom., 8(3):315-334, 1992. ACM Symposium on Computational Geometry (North Conway, NH, 1991). URL: https://doi.org/10.1007/BF02293051.
  60. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete Comput. Geom., 10(2):157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
  61. Jiří Matoušek and Zuzana Patáková. Multilevel polynomial partitions and simplified range searching. Discrete Comput. Geom., 54(1):22-41, 2015. URL: https://doi.org/10.1007/s00454-015-9701-2.
  62. Joseph O'Rourke. Visibility. In Handbook of Discrete and Computational Geometry, 3rd Edition, chapter 33. Chapman and Hall/CRC, 2017. URL: https://doi.org/10.1201/9781420035315.CH28.
  63. János Pach and Micha Sharir. Incidences. In Graph theory, combinatorics and algorithms, pages 267-292. Springer, New York, 2005. URL: https://doi.org/10.1007/0-387-25036-0_11.
  64. Larry Palazzi and Jack Snoeyink. Counting and reporting red/blue segment intersections. CVGIP: Graphical Models and Image Processing, 56(4):304-310, 1994. URL: https://doi.org/10.1006/cgip.1994.1027.
  65. Franco P. Preparata and Michael Ian Shamos. Computational geometry. Texts and Monographs in Computer Science. Springer-Verlag, New York, 1985. URL: https://doi.org/10.1007/978-1-4612-1098-6.
  66. Vojtech Rödl and Andrzej Ruciński. Bipartite coverings of graphs. Combinatorics, Probability and Computing, 6(3):349-352, 1997. URL: https://doi.org/10.1017/S0963548397003064.
  67. Lisa Sauermann. On the speed of algebraically defined graph classes. Advances in Mathematics, 380:107593, 2021. URL: https://doi.org/10.1016/j.aim.2021.107593.
  68. Stephan Schwartz. An overview of graph covering and partitioning. Discrete Math., 345(8):Paper No. 112884, 17, 2022. URL: https://doi.org/10.1016/j.disc.2022.112884.
  69. Anish Man Singh Shrestha, Satoshi Tayu, and Shuichi Ueno. On orthogonal ray graphs. Discrete Appl. Math., 158(15):1650-1659, 2010. URL: https://doi.org/10.1016/j.dam.2010.06.002.
  70. Shakhar Smorodinsky. A survey of Zarankiewicz problems in geometry. arXiv preprint arXiv:2410.03702, 2024. URL: https://doi.org/10.48550/arXiv.2410.03702.
  71. Andrew Suk. Semi-algebraic Ramsey numbers. J. Combin. Theory Ser. B, 116:465-483, 2016. URL: https://doi.org/10.1016/j.jctb.2015.10.001.
  72. Xiaoying Tang, Li Xia, Yezi Liao, Weifeng Liu, Yuhua Peng, Tianxin Gao, and Yanjun Zeng. New approach to epileptic diagnosis using visibility graph of high-frequency signal. Clinical EEG and Neuroscience, 44(2):150-156, 2013. PMID: 23508995. URL: https://doi.org/10.1177/1550059412464449.
  73. István Tomon. Ramsey properties of semilinear graphs. Israel Journal of Mathematics, 254(1):113-139, 2023. URL: https://doi.org/10.1007/s11856-022-2390-7.
  74. Csaba D. Tóth. Weighted biclique covers. Open problem posed at the Tenth Annual Workshop on Geometry and Graphs (WoGaG'23), Bellairs Research Institute, 2023. Google Scholar
  75. Zsolt Tuza. Covering of graphs by complete bipartite subgraphs: complexity of 0-1 matrices. Combinatorica, 4(1):111-116, 1984. URL: https://doi.org/10.1007/BF02579163.
  76. Jan van den Brand, Li Chen, Richard Peng, Rasmus Kyng, Yang P. Liu, Maximilian Probst Gutenberg, Sushant Sachdeva, and Aaron Sidford. A deterministic almost-linear time algorithm for minimum-cost flow. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 503-514. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00037.
  77. Peter van Emde Boas. Preserving order in a forest in less than logarithmic time. In 16th Annual Symposium on Foundations of Computer Science (Univ. California, Berkeley, Calif., 1975), pages 75-84. IEEE, Long Beach, CA, 1975. URL: https://doi.org/10.1109/SFCS.1975.26.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact