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Upward Book Embeddings of st-Graphs

Authors Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, Maurizio Patrignani

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Author Details

Carla Binucci
  • Università degli Studi di Perugia, Perugia, Italy
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Emilio Di Giacomo
  • Università degli Studi di Perugia, Perugia, Italy
Walter Didimo
  • Università degli Studi di Perugia, Perugia, Italy
Tamara Mchedlidze
  • Karlsruhe Institute of Technology, Karlsruhe, Germany
Maurizio Patrignani
  • Roma Tre University, Rome, Italy

Funding

This work was supported in part by project "Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni" - Ricerca di Base 2018, Dipartimento di Ingegneria dell'Università degli Studi di Perugia (Binucci, Di Giacomo, and Didimo), and in part by MIUR Project "MODE" under PRIN 20157EFM5C, by MIUR Project "AHeAD" under PRIN 20174LF3T8, by MIUR-DAAD JMP No 34120, by H2020-MSCA-RISE project 734922 - "CONNECT", and by Roma Tre University Azione 4 Project "GeoView" (Da Lozzo and Patrignani).

Acknowledgements

This research began at the Bertinoro Workshop on Graph Drawing 2018.

Cite As Get BibTex

Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, and Maurizio Patrignani. Upward Book Embeddings of st-Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.13

Abstract

We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • Upward Book Embeddings
  • st-Graphs
  • SPQR-trees
  • Branchwidth
  • Sphere-cut Decomposition

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Related Versions

  • Details for the omitted and sketched proofs can be found in the full version of the paper [Carla Binucci et al., 2019], which is available at https://arxiv.org/abs/1903.07966.

References

  1. Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, and Gelasio Salazar. The 2-page crossing number of K_n. In Tamal K. Dey and Sue Whitesides, editors, Symposium on Computational Geometry 2012, SoCG '12, pages 397-404. ACM, 2012. URL: http://dx.doi.org/10.1145/2261250.2261310.
  2. Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, and Quanquan C. Liu. Upward Partitioned Book Embeddings. In Fabrizio Frati and Kwan-Liu Ma, editors, GD 2017, volume 10692 of LNCS, pages 210-223. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-73915-1_18.
  3. Mustafa Alhashem, Guy-Vincent Jourdan, and Nejib Zaguia. On The Book Embedding Of Ordered Sets. Ars Combinatoria, 119:47-64, 2015. Google Scholar
  4. Mohammad Alzohairi and Ivan Rival. Series-Parallel Planar Ordered Sets Have Pagenumber Two. In Stephen C. North, editor, Graph Drawing, GD '96, volume 1190 of LNCS, pages 11-24. Springer, 1996. URL: http://dx.doi.org/10.1007/3-540-62495-3_34.
  5. Patrizio Angelini, Michael A. Bekos, Walter Didimo, Luca Grilli, Philipp Kindermann, Tamara Mchedlidze, Roman Prutkin, Antonios Symvonis, and Alessandra Tappini. Greedy Rectilinear Drawings. In Therese Biedl and Andreas Kerren, editors, GD 2018, volume 11282 of LNCS. Springer, 2018. Google Scholar
  6. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Valentino Di Donato, Philipp Kindermann, Günter Rote, and Ignaz Rutter. Windrose Planarity: Embedding Graphs with Direction-Constrained Edges. ACM Trans. Algorithms, 14(4):54:1-54:24, 2018. URL: http://dx.doi.org/10.1145/3239561.
  7. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, and Fabrizio Frati. Strip Planarity Testing for Embedded Planar Graphs. Algorithmica, 77(4):1022-1059, 2017. URL: http://dx.doi.org/10.1007/s00453-016-0128-9.
  8. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. Intersection-Link Representations of Graphs. J. Graph Algorithms Appl., 21(4):731-755, 2017. URL: http://dx.doi.org/10.7155/jgaa.00437.
  9. Patrizio Angelini, Giordano Da Lozzo, and Daniel Neuwirth. Advancements on SEFE and Partitioned Book Embedding problems. Theor. Comput. Sci., 575:71-89, 2015. Google Scholar
  10. Patrizio Angelini, Marco Di Bartolomeo, and Giuseppe Di Battista. Implementing a Partitioned 2-Page Book Embedding Testing Algorithm. In Graph Drawing, volume 7704 of LNCS, pages 79-89. Springer, 2012. Google Scholar
  11. Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph. J. Discrete Algorithms, 14:150-172, 2012. Google Scholar
  12. Patrizio Angelini, David Eppstein, Fabrizio Frati, Michael Kaufmann, Sylvain Lazard, Tamara Mchedlidze, Monique Teillaud, and Alexander Wolff. Universal Point Sets for Drawing Planar Graphs with Circular Arcs. J. Graph Algorithms Appl., 18(3):313-324, 2014. Google Scholar
  13. Melanie Badent, Emilio Di Giacomo, and Giuseppe Liotta. Drawing colored graphs on colored points. Theor. Comput. Sci., 408(2-3):129-142, 2008. Google Scholar
  14. Michael J. Bannister and David Eppstein. Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth. In Christian A. Duncan and Antonios Symvonis, editors, GD 2014, volume 8871 of LNCS, pages 210-221. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-45803-7_18.
  15. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999. Google Scholar
  16. 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, 2017. Google Scholar
  17. Michael A. Bekos, Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Chrysanthi N. Raftopoulou. Edge Partitions of Optimal 2-plane and 3-plane Graphs. In Andreas Brandstädt, Ekkehard Köhler, and Klaus Meer, editors, Graph-Theoretic Concepts in Computer Science, WG 2018, volume 11159 of LNCS, pages 27-39. Springer, 2018. URL: http://dx.doi.org/10.1007/978-3-030-00256-5_3.
  18. Michael A. Bekos, Martin Gronemann, and Chrysanthi N. Raftopoulou. Two-Page Book Embeddings of 4-Planar Graphs. Algorithmica, 75(1):158-185, 2016. URL: http://dx.doi.org/10.1007/s00453-015-0016-8.
  19. Frank Bernhart and Paul C Kainen. The book thickness of a graph. Journal of Combinatorial Theory, Series B, 27(3):320-331, 1979. URL: http://dx.doi.org/10.1016/0095-8956(79)90021-2.
  20. Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-Upward Planarity. Algorithmica, 32(3):474-506, 2002. URL: http://dx.doi.org/10.1007/s00453-001-0083-x.
  21. Paola Bertolazzi, Giuseppe Di Battista, Carlo Mannino, and Roberto Tamassia. Optimal Upward Planarity Testing of Single-Source Digraphs. SIAM J. Comput., 27(1):132-169, 1998. URL: http://dx.doi.org/10.1137/S0097539794279626.
  22. Therese C. Biedl, Thomas C. Shermer, Sue Whitesides, and Stephen K. Wismath. Bounds for Orthogonal 3-D Graph Drawing. J. Graph Algorithms Appl., 3(4):63-79, 1999. Google Scholar
  23. Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, and Maurizio Patrignani. Upward Book Embeddings of st-Graphs. CoRR, abs/1903.07966, 2019. URL: http://arxiv.org/abs/1903.07966.
  24. Carla Binucci, Emilio Di Giacomo, Md. Iqbal Hossain, and Giuseppe Liotta. 1-page and 2-page drawings with bounded number of crossings per edge. Eur. J. Comb., 68:24-37, 2018. URL: http://dx.doi.org/10.1016/j.ejc.2017.07.009.
  25. Carla Binucci and Walter Didimo. Computing Quasi-Upward Planar Drawings of Mixed Graphs. Comput. J., 59(1):133-150, 2016. URL: http://dx.doi.org/10.1093/comjnl/bxv082.
  26. Franz-Josef Brandenburg. Upward planar drawings on the standing and the rolling cylinders. Comput. Geom., 47(1):25-41, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2013.08.003.
  27. Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc diagrams, flip distances, and Hamiltonian triangulations. Comput. Geom., 68:206-225, 2018. URL: http://dx.doi.org/10.1016/j.comgeo.2017.06.001.
  28. Steven Chaplick, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Martin Nöllenburg, Maurizio Patrignani, Ioannis G. Tollis, and Alexander Wolff. Planar L-Drawings of Directed Graphs. In Fabrizio Frati and Kwan-Liu Ma, editors, GD 2017, volume 10692 of LNCS, pages 465-478. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-73915-1_36.
  29. Fan R. K. Chung, Frank Thomson Leighton, and Arnold L. Rosenberg. Embedding graphs in books: A layout problem with applications to VLSI design. SIAM Journal on Algebraic Discrete Methods, 8(1):33-58, 1987. Google Scholar
  30. Robert J. Cimikowski. An analysis of some linear graph layout heuristics. J. Heuristics, 12(3):143-153, 2006. Google Scholar
  31. Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. Upward Planar Morphs. In Therese C. Biedl and Andreas Kerren, editors, GD 2018, volume 11282 of LNCS, pages 92-105. Springer, 2018. URL: http://dx.doi.org/10.1007/978-3-030-04414-5_7.
  32. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall PTR, Upper Saddle River, NJ, USA, 1998. Google Scholar
  33. Giuseppe Di Battista and Roberto Tamassia. Algorithms for Plane Representations of Acyclic Digraphs. Theor. Comput. Sci., 61:175-198, 1988. URL: http://dx.doi.org/10.1016/0304-3975(88)90123-5.
  34. Giuseppe Di Battista and Roberto Tamassia. On-Line Planarity Testing. SIAM Journal on Computing, 25(5):956-997, 1996. Google Scholar
  35. Giuseppe Di Battista, Roberto Tamassia, and Ioannis G. Tollis. Area Requirement and Symmetry Display of Planar Upward Drawings. Discrete & Computational Geometry, 7:381-401, 1992. URL: http://dx.doi.org/10.1007/BF02187850.
  36. Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Stephen K. Wismath. Curve-constrained drawings of planar graphs. Comput. Geom., 30(1):1-23, 2005. URL: http://dx.doi.org/10.1016/j.comgeo.2004.04.002.
  37. Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Stephen K. Wismath. Book Embeddability of Series-Parallel Digraphs. Algorithmica, 45(4):531-547, 2006. URL: http://dx.doi.org/10.1007/s00453-005-1185-7.
  38. Emilio Di Giacomo, Francesco Giordano, and Giuseppe Liotta. Upward Topological Book Embeddings of DAGs. SIAM Journal on Discrete Mathematics, 25(2):479-489, 2011. URL: http://dx.doi.org/10.1137/080731128.
  39. Emilio Di Giacomo, Giuseppe Liotta, and Francesco Trotta. On Embedding a Graph on Two Sets of Points. Int. J. Found. Comput. Sci., 17(5):1071-1094, 2006. Google Scholar
  40. Emilio Di Giacomo, Giuseppe Liotta, and Francesco Trotta. Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge. Algorithmica, 57(4):796-818, 2010. Google Scholar
  41. Walter Didimo, Francesco Giordano, and Giuseppe Liotta. Upward Spirality and Upward Planarity Testing. SIAM J. Discrete Math., 23(4):1842-1899, 2009. Google Scholar
  42. Vida Dujmović and David R. Wood. On Linear Layouts of Graphs. Discrete Mathematics & Theoretical Computer Science, 6(2):339-358, 2004. Google Scholar
  43. Vida Dujmović and David R. Wood. Stacks, Queues and Tracks: Layouts of Graph Subdivisions. Discrete Mathematics & Theoretical Computer Science, 7(1):155-202, 2005. URL: http://dmtcs.episciences.org/346.
  44. H. Enomoto and M. S. Miyauchi. Embedding Graphs into a Three Page Book with O(m log n) Crossings of Edges over the Spine. SIAM J. Discrete Math., 12(3):337-341, 1999. Google Scholar
  45. H. Enomoto, M. S. Miyauchi, and K. Ota. Lower Bounds for the Number of Edge-crossings Over the Spine in a Topological Book Embedding of a Graph. Discrete Applied Mathematics, 92(2-3):149-155, 1999. Google Scholar
  46. Hikoe Enomoto, Tomoki Nakamigawa, and Katsuhiro Ota. On the Pagenumber of Complete Bipartite Graphs. Journal of Combinatorial Theory, Series B, 71(1):111-120, 1997. URL: http://dx.doi.org/10.1006/jctb.1997.1773.
  47. Hazel Everett, Sylvain Lazard, Giuseppe Liotta, and Stephen K. Wismath. Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices. Discrete & Computational Geometry, 43(2):272-288, 2010. Google Scholar
  48. Fedor V. Fomin and Dimitrios M. Thilikos. New upper bounds on the decomposability of planar graphs. Journal of Graph Theory, 51(1):53-81, 2006. URL: http://dx.doi.org/10.1002/jgt.20121.
  49. Fabrizio Frati, Radoslav Fulek, and Andres J. Ruiz-Vargas. On the Page Number of Upward Planar Directed Acyclic Graphs. Journal of Graph Algorithms and Applications, 17(3):221-244, 2013. URL: http://dx.doi.org/10.7155/jgaa.00292.
  50. Joseph L. Ganley and Lenwood S. Heath. The pagenumber of k-trees is O(k). Discrete Applied Mathematics, 109(3):215-221, 2001. Google Scholar
  51. Ashim Garg and Roberto Tamassia. On the Computational Complexity of Upward and Rectilinear Planarity Testing. SIAM J. Comput., 31(2):601-625, 2001. URL: http://dx.doi.org/10.1137/S0097539794277123.
  52. Francesco Giordano, Giuseppe Liotta, Tamara Mchedlidze, Antonios Symvonis, and Sue Whitesides. Computing upward topological book embeddings of upward planar digraphs. Journal of Discrete Algorithms, 30:45-69, 2015. Google Scholar
  53. L. Heath, F. Leighton, and A. Rosenberg. Comparing Queues and Stacks As Mechanisms for Laying Out Graphs. SIAM Journal on Discrete Mathematics, 5(3):398-412, 1992. Google Scholar
  54. Lenwood S. Heath and Sriram V. Pemmaraju. Stack and Queue Layouts of Posets. SIAM Journal on Discrete Mathematics, 10(4):599-625, 1997. Google Scholar
  55. Lenwood S. Heath and Sriram V. Pemmaraju. Stack and Queue Layouts of Directed Acyclic Graphs: Part II. SIAM Journal on Computing, 28(5):1588-1626, 1999. Google Scholar
  56. 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, 1999. Google Scholar
  57. Seok-Hee Hong and Hiroshi Nagamochi. Simpler algorithms for testing two-page book embedding of partitioned graphs. Theor. Comput. Sci., 725:79-98, 2018. Google Scholar
  58. L. T. Q. Hung. A Planar Poset which Requires 4 Pages. PhD thesis, Institute of Computer Science, University of Wrocław, 1989. Google Scholar
  59. Maarten Löffler and Csaba D. Tóth. Linear-Size Universal Point Sets for One-Bend Drawings. In Graph Drawing, volume 9411 of LNCS, pages 423-429. Springer, 2015. Google Scholar
  60. Seth M. Malitz. Genus g Graphs Have Pagenumber O(√g). J. Algorithms, 17(1):85-109, 1994. Google Scholar
  61. Seth M. Malitz. Graphs with E Edges Have Pagenumber O(√E). J. Algorithms, 17(1):71-84, 1994. Google Scholar
  62. Sumio Masuda, Kazuo Nakajima, Toshinobu Kashiwabara, and Toshio Fujisawa. Crossing Minimization in Linear Embeddings of Graphs. IEEE Trans. Computers, 39(1):124-127, 1990. Google Scholar
  63. Tamara Mchedlidze and Antonios Symvonis. Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs. In Yingfei Dong, Ding-Zhu Du, and Oscar H. Ibarra, editors, Algorithms and Computation, ISAAC 2009, volume 5878 of LNCS, pages 882-891. Springer, 2009. Google Scholar
  64. Tamara Mchedlidze and Antonios Symvonis. Unilateral Orientation of Mixed Graphs. In SOFSEM 2010, volume 5901 of LNCS, pages 588-599. Springer, 2010. Google Scholar
  65. Tamara Mchedlidze and Antonios Symvonis. Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs. Journal of Graph Algorithms and Applications, 15(3):373-415, 2011. URL: http://dx.doi.org/10.7155/jgaa.00231.
  66. Richard Nowakowski and Andrew Parker. Ordered sets, pagenumbers and planarity. Order, 6(3):209-218, 1989. Google Scholar
  67. J. Opatrny. Total Ordering Problem. SIAM Journal on Computing, 8(1):111-114, 1979. URL: http://dx.doi.org/10.1137/0208008.
  68. Sriram V. Pemmaraju. Exploring the Powers of Stacks and Queues via Graph Layouts. PhD thesis, Virginia Polytechnic Institute and State University at Blacksburg, Virginia, 1992. Google Scholar
  69. Aimal Rextin and Patrick Healy. Dynamic Upward Planarity Testing of Single Source Embedded Digraphs. Comput. J., 60(1):45-59, 2017. URL: http://dx.doi.org/10.1093/comjnl/bxw064.
  70. 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: http://dx.doi.org/10.1016/0095-8956(91)90061-N.
  71. Paul D. Seymour and Robin Thomas. Call Routing and the Ratcatcher. Combinatorica, 14(2):217-241, 1994. URL: http://dx.doi.org/10.1007/BF01215352.
  72. Maciej M. Syslo. Bounds to the Page Number of Partially Ordered Sets. In Manfred Nagl, editor, Graph-Theoretic Concepts in Computer Science, WG '89, volume 411 of LNCS, pages 181-195. Springer, 1989. URL: http://dx.doi.org/10.1007/3-540-52292-1_13.
  73. Walter Unger. On the k-Colouring of Circle-Graphs. In Robert Cori and Martin Wirsing, editors, STACS 88, volume 294 of LNCS, pages 61-72. Springer, 1988. URL: http://dx.doi.org/10.1007/BFb0035832.
  74. Walter Unger. The Complexity of Colouring Circle Graphs (Extended Abstract). In Alain Finkel and Matthias Jantzen, editors, STACS 92, volume 577 of LNCS, pages 389-400. Springer, 1992. URL: http://dx.doi.org/10.1007/3-540-55210-3_199.
  75. Avi Wigderson. The complexity of the Hamiltonian circuit problem for maximal planar graphs. Technical report, 298, EECS Department, Princeton University, 1982. Google Scholar
  76. David R. Wood. Bounded Degree Book Embeddings and Three-Dimensional Orthogonal Graph Drawing. In Graph Drawing, volume 2265 of LNCS, pages 312-327. Springer, 2001. Google Scholar
  77. Mihalis Yannakakis. Embedding Planar Graphs in Four Pages. Journal of Computer and System Sciences, 38(1):36-67, 1989. URL: http://dx.doi.org/10.1016/0022-0000(89)90032-9.
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