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Forbidden Patterns in Mixed Linear Layouts

Authors Deborah Haun , Laura Merker , Sergey Pupyrev

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Author Details

Deborah Haun
  • Karlsruhe Institute of Technology, Germany
Laura Merker
  • Karlsruhe Institute of Technology, Germany
Sergey Pupyrev
  • Menlo Park, CA, USA

Funding

  • Haun, Deborah: supported by the Deutsche Forschungsgemeinschaft - 520723789.

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Deborah Haun, Laura Merker, and Sergey Pupyrev. Forbidden Patterns in Mixed Linear Layouts. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 45:1-45:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.45

Abstract

An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue) number of an ordered graph is the minimum number of required stacks (queues). Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. The minimum number of stacks plus queues is called the mixed page number. It is well known that ordered graphs with small stack number are characterized, up to a function, by the absence of large twists (that is, pairwise crossing edges). Similarly, ordered graphs with small queue number are characterized by the absence of large rainbows (that is, pairwise nesting edges). However, no such characterization via forbidden patterns is known for mixed linear layouts.
We address this gap by introducing patterns similar to twists and rainbows, which we call thick patterns; such patterns allow a characterization, again up to a function, of mixed linear layouts of bounded-degree graphs. That is, we show that a family of ordered graphs with bounded maximum degree has bounded mixed page number if and only if the size of the largest thick pattern is bounded. In addition, we investigate an exact characterization of ordered graphs whose mixed page number equals a fixed integer k via a finite set of forbidden patterns. We show that for k = 2, there is no such characterization, which supports the nature of our first result.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Ordered Graphs
  • linear Layout
  • mixed linear Layout
  • Stack Layout
  • Queue Layout

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References

  1. A. A. Ageev. A triangle-free circle graph with chromatic number 5. Discrete Mathematics, 152(1-3):295-298, May 1996. URL: https://doi.org/10.1016/0012-365X(95)00349-2.
  2. Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. Queue layouts of planar 3-trees. Algorithmica, 82(9):2564-2585, September 2020. URL: https://doi.org/10.1007/s00453-020-00697-4.
  3. Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. The mixed page number of graphs. Theoretical Computer Science, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.036.
  4. Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. Lazy queue layouts of posets. Algorithmica, 85(5):1176-1201, May 2023. URL: https://doi.org/10.1007/s00453-022-01067-y.
  5. Patrizio Angelini, Michael A. Bekos, Philipp Kindermann, and Tamara Mchedlidze. On mixed linear layouts of series-parallel graphs. Theoretical Computer Science, 936:129-138, November 2022. URL: https://doi.org/10.1016/j.tcs.2022.09.019.
  6. Patrizio Angelini, Giordano Da Lozzo, Henry Förster, and Thomas Schneck. 2-layer k-planar graphs: Density, crossing lemma, relationships, and pathwidth. In Graph Drawing and Network Visualization: 28th International Symposium, GD 2020, pages 403-419, 2020. URL: https://doi.org/10.1007/978-3-030-68766-3_32.
  7. Michael J Bannister, William E Devanny, Vida Dujmović, David Eppstein, and David R Wood. Track layouts, layered path decompositions, and leveled planarity. Algorithmica, 81:1561-1583, 2019. URL: https://doi.org/10.1007/s00453-018-0487-5.
  8. Michael Bekos, Martin Gronemann, and Chrysanthi N. Raftopoulou. An improved upper bound on the queue number of planar graphs. Algorithmica, 85(2):544-562, February 2023. URL: https://doi.org/10.1007/s00453-022-01037-4.
  9. Michael Bekos, Michael Kaufmann, Fabian Klute, Sergey Pupyrev, Chrysanthi Raftopoulou, and Torsten Ueckerdt. Four pages are indeed necessary for planar graphs. Journal of Computational Geometry, pages 332-353 Pages, August 2020. URL: https://doi.org/10.20382/JOCG.V11I1A12.
  10. Michael A. Bekos, Till Bruckdorfer, Michael Kaufmann, and Chrysanthi N. Raftopoulou. The book thickness of 1-planar graphs is constant. Algorithmica, 79(2):444-465, October 2017. URL: https://doi.org/10.1007/s00453-016-0203-2.
  11. Michael A. Bekos, Henry Förster, Martin Gronemann, Tamara Mchedlidze, Fabrizio Montecchiani, Chrysanthi Raftopoulou, and Torsten Ueckerdt. Planar graphs of bounded degree have bounded queue number. SIAM Journal on Computing, 48(5):1487-1502, January 2019. URL: https://doi.org/10.1137/19M125340X.
  12. Frank Bernhart and Paul C Kainen. The book thickness of a graph. Journal of Combinatorial Theory, Series B, 27(3):320-331, December 1979. URL: https://doi.org/10.1016/0095-8956(79)90021-2.
  13. Sujoy Bhore, Giordano Da Lozzo, Fabrizio Montecchiani, and Martin Nöllenburg. On the upward book thickness problem: Combinatorial and complexity results. European Journal of Combinatorics, 110:103662, 2023. URL: https://doi.org/10.1016/j.ejc.2022.103662.
  14. Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, and Maurizio Patrignani. Upward book embeddability of st-graphs: Complexity and algorithms. Algorithmica, 85(12):3521-3571, 2023. URL: https://doi.org/10.1007/S00453-023-01142-Y.
  15. Jakub Černỳ. Coloring circle graphs. Electronic notes in Discrete mathematics, 29:457-461, 2007. URL: https://doi.org/10.1016/j.endm.2007.07.072.
  16. Sabine Cornelsen, Thomas Schank, and Dorothea Wagner. Drawing graphs on two and three lines. Journal of Graph Algorithms and Applications, 8(2):161-177, January 2004. URL: https://doi.org/10.7155/jgaa.00087.
  17. James Davies. Improved bounds for colouring circle graphs. Proceedings of the American Mathematical Society, July 2022. URL: https://doi.org/10.1090/proc/16044.
  18. James Davies and Rose McCarty. Circle graphs are quadratically χ-bounded. Bulletin of the London Mathematical Society, 53(3):673-679, 2021. URL: https://doi.org/10.1112/blms.12447.
  19. Philipp de Col, Fabian Klute, and Martin Nöllenburg. Mixed linear layouts: Complexity, heuristics, and experiments. In Graph Drawing and Network Visualization: 27th International Symposium, GD 2019, pages 460-467, Berlin, Heidelberg, 2019. Springer-Verlag. URL: https://doi.org/10.1007/978-3-030-35802-0_35.
  20. H. de Fraysseix, P. O. de Mendez, and J. Pach. A left-first search algorithm for planar graphs. Discrete & Computational Geometry, 13(3):459-468, June 1995. URL: https://doi.org/10.1007/BF02574056.
  21. Emilio Di Giacomo, Walter Didimo, Peter Eades, and Giuseppe Liotta. 2-layer right angle crossing drawings. Algorithmica, 68:954-997, January 2014. URL: https://doi.org/10.1007/S00453-012-9706-7.
  22. Vida Dujmović, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, and David R Wood. Planar graphs have bounded queue-number. Journal of the ACM (JACM), 67(4):1-38, 2020. URL: https://doi.org/10.1145/3385731.
  23. Vida Dujmović, Attila Pór, and David R Wood. Track layouts of graphs. Discrete Mathematics & Theoretical Computer Science, 6(2):497-522, 2004. URL: https://doi.org/10.46298/dmtcs.315.
  24. Vida Dujmović and David R Wood. Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Mathematics and Theoretical Computer Science, 7:155-202, 2005. URL: https://doi.org/10.46298/dmtcs.346.
  25. Vida Dujmović, Pat Morin, and David R. Wood. Queue layouts of graphs with bounded degree and bounded genus, 2019. https://arxiv.org/abs/1901.05594, URL: https://doi.org/10.48550/arXiv.1901.05594.
  26. Vida Dujmović, Pat Morin, and Céline Yelle. Two results on layered pathwidth and linear layouts. Journal of Graph Algorithms and Applications, 25(1):43-57, 2021. URL: https://doi.org/10.7155/jgaa.00549.
  27. Vida Dujmović and David R. Wood. On linear layouts of graphs. Discrete Mathematics & Theoretical Computer Science, Vol. 6 no. 2:317, 2004. URL: https://doi.org/10.46298/dmtcs.317.
  28. Peter Eades and Sue Whitesides. Drawing graphs in two layers. Theoretical Computer Science, 131(2):361-374, 1994. URL: https://doi.org/10.1016/0304-3975(94)90179-1.
  29. Peter Eades and Nicholas C Wormald. Edge crossings in drawings of bipartite graphs. Algorithmica, 11(4):379-403, 1994. URL: https://doi.org/10.1007/BF01187020.
  30. Martin J. Erickson. Introduction to Combinatorics. John Wiley & Sons, Ltd, 1996. URL: https://doi.org/10.1002/9781118032640.
  31. Tomás Feder and Pavol Hell. Matrix partitions of perfect graphs. Discrete Mathematics, 306(19-20):2450-2460, 2006. URL: https://doi.org/10.1016/j.disc.2005.12.035.
  32. Stefan Felsner, Torsten Ueckerdt, and Kaja Wille. On the queue-number of partial orders. In Helen C. Purchase and Ignaz Rutter, editors, Graph Drawing and Network Visualization: 29th International Symposium, GD 2021, pages 231-241, Cham, 2021. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-92931-2_17.
  33. Stefan Felsner and Lorenz Wernisch. Maximum k-chains in planar point sets: Combinatorial structure and algorithms. SIAM Journal on Computing, 28(1):192-209, 1998. URL: https://doi.org/10.1137/S0097539794266171.
  34. Fabrizio Frati, Radoslav Fulek, and Andres Ruiz-Vargas. On the page number of upward planar directed acyclic graphs. Journal of Graph Algorithms and Applications, 17(3):221-244, March 2013. URL: https://doi.org/10.7155/jgaa.00292.
  35. Henry Förster, Michael Kaufmann, Laura Merker, Sergey Pupyrev, and Chrysanthi Raftopoulou. Linear layouts of bipartite planar graphs. In Pat Morin and Subhash Suri, editors, Algorithms and Data Structures, pages 444-459, Cham, 2023. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-38906-1_29.
  36. Zoltán Füredi and Péter Hajnal. Davenport-Schinzel theory of matrices. Discrete Mathematics, 103(3):233-251, May 1992. URL: https://doi.org/10.1016/0012-365X(92)90316-8.
  37. M. R. Garey, D. S. Johnson, G. L. Miller, and C. H. Papadimitriou. The complexity of coloring circular arcs and chords. SIAM Journal on Algebraic Discrete Methods, 1(2):216-227, 1980. URL: https://doi.org/10.1137/0601025.
  38. Jesse Geneson. Almost all permutation matrices have bounded saturation functions. The Electronic Journal of Combinatorics, 28(P2.16), May 2021. URL: https://doi.org/10.37236/10124.
  39. Curtis Greene. Some partitions associated with a partially ordered set. Journal of Combinatorial Theory, Series A, 20(1):69-79, 1976. URL: https://doi.org/10.1016/0097-3165(76)90078-9.
  40. A. Gyárfás. On the chromatic number of multiple interval graphs and overlap graphs. Discrete Mathematics, 55(2):161-166, July 1985. URL: https://doi.org/10.1016/0012-365X(85)90044-5.
  41. Deborah Haun, Laura Merker, and Sergey Pupyrev. Forbidden patterns in mixed linear layouts, 2024. URL: https://doi.org/10.48550/arXiv.2412.12786.
  42. Lenwood S. Heath and Sorin Istrail. The pagenumber of genus g graphs is o(g). J. ACM, 39(3):479-501, July 1992. URL: https://doi.org/10.1145/146637.146643.
  43. Lenwood S. Heath, Sriram V. Pemmaraju, and Ann N. Trenk. Stack and queue layouts of directed acyclic graphs: Part i. SIAM Journal on Computing, 28(4):1510-1539, January 1999. URL: https://doi.org/10.1137/S0097539795280287.
  44. Lenwood S Heath and Arnold L Rosenberg. Laying out graphs using queues. SIAM Journal on Computing, 21(5):927-958, 1992. URL: https://doi.org/10.1137/0221055.
  45. Robert Hickingbotham and David R. Wood. Shallow minors, graph products, and beyond-planar graphs. SIAM Journal on Discrete Mathematics, 38(1):1057-1089, 2024. URL: https://doi.org/10.1137/22M1540296.
  46. Le Tu Quoc Hung. A planar poset which requires 4 pages. Ars Combinatoria, 35:291-302, 1993. Google Scholar
  47. Barnabás Janzer, Oliver Janzer, Van Magnan, and Abhishek Methuku. Tight general bounds for the extremal numbers of 0–1 matrices. International Mathematics Research Notices, 2024(15):11455-11463, June 2024. URL: https://doi.org/10.1093/imrn/rnae129.
  48. Paul Jungeblut, Laura Merker, and Torsten Ueckerdt. Directed acyclic outerplanar graphs have constant stack number. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1937-1952, November 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00118.
  49. Paul Jungeblut, Laura Merker, and Torsten Ueckerdt. A sublinear bound on the page number of upward planar graphs. SIAM Journal on Discrete Mathematics, 37(4):2312-2331, 2023. URL: https://doi.org/10.1137/22M1522450.
  50. Paul C. Kainen. Some recent results in topological graph theory. In Ruth A. Bari and Frank Harary, editors, Graphs and Combinatorics, pages 76-108, Berlin, Heidelberg, 1974. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/BFb0066436.
  51. Julia Katheder, Michael Kaufmann, Sergey Pupyrev, and Torsten Ueckerdt. Transforming stacks into queues: Mixed and separated layouts of graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025), 2024. URL: https://doi.org/10.48550/arXiv.2409.17776.
  52. André E Kézdy, Hunter S Snevily, and Chi Wang. Partitioning permutations into increasing and decreasing subsequences. Journal of Combinatorial Theory, Series A, 73(2):353-359, 1996. URL: https://doi.org/10.1016/S0097-3165(96)80012-4.
  53. Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. The queue-number of posets of bounded width or height. In Therese Biedl and Andreas Kerren, editors, Graph Drawing and Network Visualization: 26th International Symposium, GD 2018, pages 200-212, Cham, 2018. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-04414-5_14.
  54. Alexandr Kostochka. Upper bounds on the chromatic number of graphs. Trudy Inst. Mat.(Novosibirsk), 10(Modeli i Metody Optim.):204-226, 1988. Google Scholar
  55. Alexandr Kostochka and Jan Kratochvíl. Covering and coloring polygon-circle graphs. Discrete Mathematics, 163(1-3):299-305, 1997. URL: https://doi.org/10.1016/S0012-365X(96)00344-5.
  56. Joseph L. Ganley and Lenwood S. Heath. The pagenumber of k-trees is o(k). Discrete Applied Mathematics, 109(3):215-221, May 2001. URL: https://doi.org/10.1016/S0166-218X(00)00178-5.
  57. S. M. Malitz. Genus g graphs have pagenumber O(√g). Journal of Algorithms, 17(1):85-109, 1994. URL: https://doi.org/10.1006/jagm.1994.1028.
  58. Adam Marcus and Gábor Tardos. Excluded permutation matrices and the Stanley–Wilf conjecture. Journal of Combinatorial Theory, Series A, 107(1):153-160, July 2004. URL: https://doi.org/10.1016/j.jcta.2004.04.002.
  59. Tamara Mchedlidze and Antonios Symvonis. Crossing-free acyclic hamiltonian path completion for planar st-digraphs. In Yingfei Dong, Ding-Zhu Du, and Oscar Ibarra, editors, Algorithms and Computation (ISAAC 2009), volume 5878 of Lecture Notes in Computer Science, pages 882-891, 2009. URL: https://doi.org/10.1007/978-3-642-10631-6_89.
  60. Miki Miyauchi. Topological stack-queue mixed layouts of graphs. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E103.A(2):510-522, 2020. URL: https://doi.org/10.1587/transfun.2019EAP1097.
  61. Hiroshi Nagamochi. An improved bound on the one-sided minimum crossing number in two-layered drawings. Discrete & Computational Geometry, 33(4):569-591, 2005. URL: https://doi.org/10.1007/s00454-005-1168-0.
  62. Richard Nowakowski and Andrew Parker. Ordered sets, pagenumbers and planarity. Order, 6(3):209-218, September 1989. URL: https://doi.org/10.1007/BF00563521.
  63. Martin Nöllenburg and Sergey Pupyrev. On families of planar dags with constant stack number. In Graph Drawing and Network Visualization: 31st International Symposium, GD 2023, pages 135-151, Berlin, Heidelberg, January 2024. Springer-Verlag. URL: https://doi.org/10.1007/978-3-031-49272-3_10.
  64. L. Taylor Ollmann. On the book thicknesses of various graphs. In Proc. 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, volume 8, 1973. Google Scholar
  65. Shannon Overbay. Generalized book embeddings. Phd thesis, Colorado State University, USA, November 1998. Google Scholar
  66. János Pach and Gábor Tardos. Forbidden paths and cycles in ordered graphs and matrices. Israel Journal of Mathematics, 155:359-380, 2006. URL: https://doi.org/10.1007/BF02773960.
  67. Sriram V Pemmaraju. Exploring the powers of stacks and queues via graph layouts. PhD thesis, Virginia Tech, 1992. Google Scholar
  68. Sergey Pupyrev. Mixed linear layouts of planar graphs. In Fabrizio Frati and Kwan-Liu Ma, editors, Graph Drawing and Network Visualization: 25th International Symposium, GD 2017, pages 197-209, Cham, 2018. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-73915-1_17.
  69. Sergey Pupyrev. Queue layouts of two-dimensional posets. In Patrizio Angelini and Reinhard von Hanxleden, editors, Graph Drawing and Network Visualization: 30th International Symposium, GD 2022, pages 353-360, Cham, 2023. Springer International Publishing. URL: https://doi.org/10.1007/978-3-031-22203-0_25.
  70. Matthew Suderman. Pathwidth and layered drawings of trees. International Journal of Computational Geometry & Applications, 14(03):203-225, 2004. URL: https://doi.org/10.1142/S0218195904001433.
  71. Gábor Tardos. On 0–1 matrices and small excluded submatrices. Journal of Combinatorial Theory, Series A, 111(2):266-288, August 2005. URL: https://doi.org/10.1016/j.jcta.2004.11.015.
  72. Gábor Tardos. Extremal theory of ordered graphs. In Proceedings of the International Congress of Mathematicians (ICM 2018), pages 3235-3243, Rio de Janeiro, Brazil, May 2019. WORLD SCIENTIFIC. URL: https://doi.org/10.1142/9789813272880_0179.
  73. Alexander Tiskin. Fast RSK correspondence by doubling search. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms (ESA 2022), volume 244 of Leibniz International Proceedings in Informatics (LIPIcs), pages 86:1-86:10, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.86.
  74. Walter Unger. On the k-colouring of circle-graphs. In Robert Cori and Martin Wirsing, editors, STACS 88, pages 61-72, Berlin, Heidelberg, 1988. Springer. URL: https://doi.org/10.1007/BFb0035832.
  75. Walter Unger. The complexity of colouring circle graphs. In STACS 92: 9th Annual Symposium on Theoretical Aspects of Computer Science Cachan, France, February 13-15, 1992 Proceedings 9, pages 389-400. Springer, 1992. URL: https://doi.org/10.1007/3-540-55210-3_199.
  76. V. G. Vizing. The chromatic class of a multigraph. Cybernetics, 1(3):32-41, May 1965. URL: https://doi.org/10.1007/BF01885700.
  77. David Wärn. Partitioning permutations into monotone subsequences. Electron. J. Comb., 28(3), 2021. URL: https://doi.org/10.37236/10267.
  78. David R Wood. Bounded-degree graphs have arbitrarily large queue-number. Discrete Mathematics & Theoretical Computer Science, 10(1), 2008. URL: https://doi.org/10.46298/dmtcs.434.
  79. David R Wood. 2-layer graph drawings with bounded pathwidth. Journal of Graph Algorithms and Applications, 27(9):843-851, November 2023. URL: https://doi.org/10.7155/jgaa.00647.
  80. Mihalis Yannakakis. Embedding planar graphs in four pages. Journal of Computer and System Sciences, 38(1):36-67, February 1989. URL: https://doi.org/10.1016/0022-0000(89)90032-9.
  81. Mihalis Yannakakis. Planar graphs that need four pages. Journal of Combinatorial Theory, Series B, 145:241-263, November 2020. URL: https://doi.org/10.1016/j.jctb.2020.05.008.
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