A Tight Subexponential-Time Algorithm for Two-Page Book Embedding

Authors Robert Ganian , Haiko Müller , Sebastian Ordyniak , Giacomo Paesani , Mateusz Rychlicki

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Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Haiko Müller
  • School of Computing, University of Leeds, UK
Sebastian Ordyniak
  • School of Computing, University of Leeds, UK
Giacomo Paesani
  • School of Computing, University of Leeds, UK
Mateusz Rychlicki
  • School of Computing, University of Leeds, UK

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Robert Ganian, Haiko Müller, Sebastian Ordyniak, Giacomo Paesani, and Mateusz Rychlicki. A Tight Subexponential-Time Algorithm for Two-Page Book Embedding. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 68:1-68:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


A book embedding of a graph is a drawing that maps vertices onto a line and edges to simple pairwise non-crossing curves drawn into "pages", which are half-planes bounded by that line. Two-page book embeddings, i.e., book embeddings into 2 pages, are of special importance as they are both NP-hard to compute and have specific applications. We obtain a 2^𝒪(√n) algorithm for computing a book embedding of an n-vertex graph on two pages - a result which is asymptotically tight under the Exponential Time Hypothesis. As a key tool in our approach, we obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph. We conclude by establishing the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question arising from previous work on the problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • book embedding
  • page number
  • subexponential algorithms
  • subhamiltonicity
  • feedback edge number


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  1. Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, and Gelasio Salazar. The 2-page crossing number of K_n. Discrete & Computational Geometry, 49(4):747-777, 2013. URL: https://doi.org/10.1007/S00454-013-9514-0.
  2. Patrizio Angelini, Marco Di Bartolomeo, and Giuseppe Di Battista. Implementing a partitioned 2-page book embedding testing algorithm. Proc. GD 2012, 7704:79-89, 2012. URL: https://doi.org/10.1007/978-3-642-36763-2_8.
  3. Michael J. Bannister and David Eppstein. Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth. Journal of Graph Algorithms and Applications, 22(4):577-606, 2018. URL: https://doi.org/10.7155/jgaa.00479.
  4. Giuseppe Di Battista and Roberto Tamassia. Incremental planarity testing. Proc. FOCS 1989, pages 436-441, 1989. URL: https://doi.org/10.1109/SFCS.1989.63515.
  5. Michael A. Bekos, Martin Gronemann, and Chrysanthi N. Raftopoulou. Two-page book embeddings of 4-planar graphs. Algorithmica, 75(1):158-185, 2016. URL: https://doi.org/10.1007/s00453-015-0016-8.
  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. Sujoy Bhore, Robert Ganian, Fabrizio Montecchiani, and Martin Nöllenburg. Parameterized algorithms for book embedding problems. Journal of Graph Algorithms and Applications, 24(4):603-620, 2020. URL: https://doi.org/10.7155/jgaa.00526.
  8. Daniel Bienstock and Clyde L. Monma. Optimal enclosing regions in planar graphs. Networks, 19(1):79-94, 1989. URL: https://doi.org/10.1002/NET.3230190107.
  9. Daniel Bienstock and Clyde L. Monma. On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica, 5(1):93-109, 1990. URL: https://doi.org/10.1007/BF01840379.
  10. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. URL: https://doi.org/10.1016/J.IC.2014.12.008.
  11. F. Chung, F. Leighton, and A. Rosenberg. Embedding graphs in books: a layout problem with applications to VLSI design. SIAM Journal on Algebraic Discrete Methods, 8(1):33-58, 1987. URL: https://doi.org/10.1137/0608002.
  12. Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  13. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  14. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. Journal of the ACM, 65(3):12:1-12:46, 2018. URL: https://doi.org/10.1145/3148227.
  15. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. ACM Transactions on Algorithms, 18(2):17:1-17:31, 2022. URL: https://doi.org/10.1145/3506707.
  16. Hubert de Fraysseix, Patrice Ossona de Mendez, and János Pach. A left-first search algorithm for planar graphs. Discrete & Computational Geometry, 13:459-468, 1995. URL: https://doi.org/10.1007/BF02574056.
  17. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  18. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. URL: https://doi.org/10.1007/S00453-009-9296-1.
  19. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  20. Vida Dujmović and David R. Wood. On linear layouts of graphs. Discrete Mathematics & Theoretical Computer Science, 6(2):339-358, 2004. URL: https://doi.org/10.46298/dmtcs.317.
  21. Vida Dujmovic and David R. Wood. Graph treewidth and geometric thickness parameters. Discrete & Computational Geometry, 37(4):641-670, 2007. URL: https://doi.org/10.1007/s00454-007-1318-7.
  22. Toshiki Endo. The pagenumber of toroidal graphs is at most seven. Discrete Mathematics, 175(1):87-96, 1997. URL: https://doi.org/10.1016/S0012-365X(96)00144-6.
  23. Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, and Meirav Zehavi. Parameterized complexity in graph drawing (dagstuhl seminar 21293). Dagstuhl Reports, 11(6):82-123, 2021. URL: https://doi.org/10.1016/j.artint.2017.12.006.
  24. Robert Ganian, Haiko Mueller, Sebastian Ordyniak, Giacomo Paesani, and Mateusz Rychlicki. A tight subexponential-time algorithm for two-page book embedding, 2024. URL: https://arxiv.org/abs/2404.14087.
  25. 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.
  26. M. R. Garey, D. S. Johnson, and R. Endre Tarjan. The planar hamiltonian circuit problem is np-complete. SIAM Journal on Computing, 5(4):704-714, 1976. URL: https://doi.org/10.1137/0205049.
  27. Emilio Di Giacomo and Giuseppe Liotta. The hamiltonian augmentation problem and its applications to graph drawing. Proc. WALCOM 2010, LNCS, 5942:35-46, 2010. URL: https://doi.org/10.1007/978-3-642-11440-3_4.
  28. Qian-Ping Gu and Hisao Tamaki. Improved bounds on the planar branchwidth with respect to the largest grid minor size. Algorithmica, 64(3):416-453, 2012. URL: https://doi.org/10.1007/S00453-012-9627-5.
  29. Carsten Gutwenger, Petra Mutzel, and René Weiskircher. Inserting an edge into a planar graph. Algorithmica, 41(4):289-308, 2005. URL: https://doi.org/10.1007/S00453-004-1128-8.
  30. András Gyárfás and Jenö Lehel. Covering and coloring problems for relatives of intervals. Discrete Mathematics, 55(2):167-180, 1985. URL: https://doi.org/10.1016/0012-365X(85)90045-7.
  31. Christian Haslinger and Peter F. Stadler. RNA structures with pseudo-knots: Graph-theoretical, combinatorial, and statistical properties. Bulletin of Mathematical Biology, 61(3):437-467, 1999. URL: https://doi.org/10.1006/bulm.1998.0085.
  32. Lenwood S. Heath. Embedding outerplanar graphs in small books. SIAM Journal on Algebraic Discrete Methods, 8(2):198-218, 1987. URL: https://doi.org/10.1137/0608018.
  33. Seok-Hee Hong and Hiroshi Nagamochi. Two-page book embedding and clustered graph planarity. Technical report, Citeseer, 2009. Google Scholar
  34. Seok-Hee Hong and Hiroshi Nagamochi. Simpler algorithms for testing two-page book embedding of partitioned graphs. Theoretical Computer Science, 725:79-98, 2018. URL: https://doi.org/10.1016/J.TCS.2015.12.039.
  35. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/JCSS.2001.1774.
  36. Hugo Jacob and Marcin Pilipczuk. Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs. Proc. WG 2022, 13453:287-299, 2022. URL: https://doi.org/10.1007/978-3-031-15914-5_21.
  37. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. Proc. FOCS 2021, pages 184-192, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  38. Seth 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.
  39. Dániel Marx. Four shorts stories on surprising algorithmic uses of treewidth. Treewidth, Kernels, and Algorithms, 12160:129-144, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_10.
  40. Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. A subexponential parameterized algorithm for directed subset traveling salesman problem on planar graphs. SIAM Journal on Computing, 51(2):254-289, 2022. URL: https://doi.org/10.1137/19M1304088.
  41. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  42. Malgorzata Nowicka, Vinay K. Gautam, and Pekka Orponen. Automated rendering of multi-stranded dna complexes with pseudoknots, 2023. URL: https://arxiv.org/abs/2308.06392.
  43. Neil Robertson and Paul D. Seymour. Graph minors. x. obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B, 52(2):153-190, 1991. URL: https://doi.org/10.1016/0095-8956(91)90061-N.
  44. Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a planar graph. Journal of Combinatorial Theory, Series B, 62(2):323-348, 1994. URL: https://doi.org/10.1006/JCTB.1994.1073.
  45. Johannes Uhlmann and Mathias Weller. Two-layer planarization parameterized by feedback edge set. Theoretical Computer Science, 494:99-111, 2013. URL: https://doi.org/10.1016/J.TCS.2013.01.029.
  46. Avi Wigderson. The complexity of the hamiltonian circuit problem for maximal planar graphs. Technical Report, 1982. Google Scholar
  47. 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.