Transforming Stacks into Queues: Mixed and Separated Layouts of Graphs

Authors Julia Katheder , Michael Kaufmann , Sergey Pupyrev , Torsten Ueckerdt



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Author Details

Julia Katheder
  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany
Michael Kaufmann
  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany
Sergey Pupyrev
  • Menlo Park, CA, USA
Torsten Ueckerdt
  • Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.56

Abstract

Some of the most important open problems for linear layouts of graphs ask for the relation between a graph’s queue number and its stack number or mixed number. In such, we seek a vertex order and edge partition of G into parts with pairwise non-crossing edges (a stack) or with pairwise non-nesting edges (a queue). Allowing only stacks, only queues, or both, the minimum number of required parts is the graph’s stack number sn(G), queue number qn(G), and mixed number mn(G), respectively.
Already in 1992, Heath and Rosenberg asked whether qn(G) is bounded in terms of sn(G), that is, whether stacks "can be transformed into" queues. This is equivalent to bipartite 3-stack graphs having bounded queue number (Dujmović and Wood, 2005). Recently, Alam et al. asked whether qn(G) is bounded in terms of mn(G), which we show to also be equivalent to the previous questions.
We approach the problem by considering separated linear layouts of bipartite graphs. In this natural setting all vertices of one part must precede all vertices of the other part. Separated stack and queue numbers coincide, and for fixed vertex orders, graphs with bounded separated stack/queue number can be characterized and efficiently recognized, whereas the separated mixed layouts are more challenging. In this work, we thoroughly investigate the relationship between separated and non-separated, mixed and pure linear layouts.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorics
Keywords
  • Separated linear Layouts
  • Stack Number
  • Queue Number
  • mixed Number
  • bipartite Graphs

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References

  1. Jawaherul Md Alam, Michael A Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. Queue layouts of planar 3-trees. Algorithmica, pages 1-22, 2020. URL: https://doi.org/10.1007/s00453-020-00697-4.
  2. Patrizio Angelini, Giordano Da Lozzo, Henry Förster, and Thomas Schneck. 2-layer k-planar graphs density, crossing lemma, relationships and pathwidth. The Computer Journal, 67(3):1005-1016, April 2023. URL: https://doi.org/10.1093/comjnl/bxad038.
  3. János Barát, Jirí Matousek, and David R. Wood. Bounded-degree graphs have arbitrarily large geometric thickness. The Electronic Journal of Combinatoric, 13(1), 2006. URL: https://doi.org/10.37236/1029.
  4. Michael A. Bekos, Martin Gronemann, and Chrysanthi N. Raftopoulou. On the queue number of planar graphs. In International Symposium on Graph Drawing and Network Visualization, pages 271-284, Berlin, Heidelberg, 2021. Springer-Verlag. URL: https://doi.org/10.1007/978-3-030-92931-2_20.
  5. Michael A. Bekos, Michael Kaufmann, Fabian Klute, Sergey Pupyrev, Chrysanthi N. Raftopoulou, and Torsten Ueckerdt. Four pages are indeed necessary for planar graphs. Journal of Computational Geometry, 11(1):332-353, 2020. URL: https://doi.org/10.20382/jocg.v11i1a12.
  6. Frank Bernhart and Paul C. Kainen. The book thickness of a graph. Journal of Combinatorial Theory, Series B, 27(3):320-331, 1979. URL: https://doi.org/10.1016/0095-8956(79)90021-2.
  7. Vida Dujmović, David Eppstein, Robert Hickingbotham, Pat Morin, and David R Wood. Stack-number is not bounded by queue-number. Combinatorica, pages 1-14, 2021. URL: https://doi.org/10.1007/s00493-021-4585-7.
  8. Vida Dujmović, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, and David R Wood. Planar graphs have bounded queue-number. Journal of the ACM, 67(4):1-38, 2020. URL: https://doi.org/10.1145/3385731.
  9. Vida Dujmović, Pat Morin, and David R Wood. Layout of graphs with bounded tree-width. SIAM Journal on Computing, 34(3):553-579, 2005. URL: https://doi.org/10.1137/S0097539702416141.
  10. 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.
  11. 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.
  12. 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.
  13. David Eppstein, Robert Hickingbotham, Laura Merker, Sergey Norin, Michał T Seweryn, and David R Wood. Three-dimensional graph products with unbounded stack-number. Discrete & Computational Geometry, pages 1-28, 2023. URL: https://doi.org/10.1007/s00454-022-00478-6.
  14. Henry Förster, Michael Kaufmann, Laura Merker, Sergey Pupyrev, and Chrysanthi N. Raftopoulou. Linear layouts of bipartite planar graphs. In Pat Morin and Subhash Suri, editors, Algorithms and Data Structures - 18th International Symposium, WADS 2023, Montreal, QC, Canada, July 31 - August 2, 2023, Proceedings, volume 14079 of Lecture Notes in Computer Science, pages 444-459. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-38906-1_29.
  15. Joseph L. Ganley and Lenwood S. Heath. The pagenumber of k-trees is O(k). Discrete Applied Mathematics, 109(3):215-221, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00178-5.
  16. 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), 2024. URL: https://doi.org/10.48550/arXiv.2412.12786.
  17. Lenwood S Heath, Frank Thomson Leighton, and Arnold L Rosenberg. Comparing queues and stacks as machines for laying out graphs. SIAM Journal on Discrete Mathematics, 5(3):398-412, 1992. URL: https://doi.org/10.1137/0405031.
  18. 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.
  19. 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.
  20. 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.
  21. Julia Katheder, Michael Kaufmann, Sergey Pupyrev, and Torsten Ueckerdt. Transforming stacks into queues: Mixed and separated layouts of graphs. CoRR, abs/2409.17776, 2024. URL: https://doi.org/10.48550/arXiv.2409.17776.
  22. Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. The mixed page number of graphs. Theoretical Computer Science, 931:131-141, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.036.
  23. 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.
  24. L Taylor Ollmann. On the book thicknesses of various graphs. In Proc. 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, volume 8, page 459. Utilitas Math., 1973. Google Scholar
  25. Sriram Venkata Pemmaraju. Exploring the powers of stacks and queues via graph layouts. PhD thesis, Virginia Polytechnic Institute and State University, 1992. Google Scholar
  26. Sergey Pupyrev. Mixed linear layouts of planar graphs. In International Symposium on Graph Drawing and Network Visualization, pages 197-209. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-73915-1_17.
  27. Sergey Pupyrev. Improved bounds for track numbers of planar graphs. Journal of Graph Algorithms and Applications, 24(3):323-341, 2020. URL: https://doi.org/10.7155/JGAA.00536.
  28. Veit Wiechert. On the queue-number of graphs with bounded tree-width. The Electronic Journal of Combinatorics, 24(1):P1.65, 2017. URL: https://doi.org/10.37236/6429.
  29. David R. Wood. Bounded-degree graphs have arbitrarily large queue-number. Discrete Mathematics and Theoretical Computer Science, 10(1), 2008. URL: https://doi.org/10.46298/DMTCS.434.
  30. Mihalis Yannakakis. Embedding planar graphs in four pages. Journal of Computer and System Sciences, 38(1):36-67, 1989. URL: https://doi.org/10.1016/0022-0000(89)90032-9.
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