Untangling Circular Drawings: Algorithms and Complexity

Authors Sujoy Bhore , Guangping Li , Martin Nöllenburg , Ignaz Rutter , Hsiang-Yun Wu



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Sujoy Bhore
  • Indian Institute of Science Education and Research, Bhopal, India
Guangping Li
  • Algorithms and Complexity Group, TU Wien, Austria
Martin Nöllenburg
  • Algorithms and Complexity Group, TU Wien, Austria
Ignaz Rutter
  • Universität Passau, Germany
Hsiang-Yun Wu
  • Research Unit of Computer Graphics, TU Wien, Austria

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Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, and Hsiang-Yun Wu. Untangling Circular Drawings: Algorithms and Complexity. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.19

Abstract

We consider the problem of untangling a given (non-planar) straight-line circular drawing δ_G of an outerplanar graph G = (V,E) into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift°(δ_G) as the minimum number of vertices that need to be shifted to resolve all crossings of δ_G. We show that the problem Circular Untangling, asking whether shift°(δ_G) ≤ K for a given integer K, is NP-complete. Based on this result we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case we provide a tight upper bound shift°(δ_G) ≤ ⌊n/2⌋-1, where n is the number of vertices in G, and present a polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Mathematics of computing → Permutations and combinations
  • Theory of computation → Problems, reductions and completeness
Keywords
  • graph drawing
  • straight-line drawing
  • outerplanarity
  • NP-hardness
  • untangling

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References

  1. Jasine Babu, Areej Khoury, and Ilan Newman. Every property of outerplanar graphs is testable. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, volume 60 of LIPIcs, pages 21:1-21:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.21.
  2. Fabian Beck, Michael Burch, Stephan Diehl, and Daniel Weiskopf. The state of the art in visualizing dynamic graphs. In 16th Eurographics Conference on Visualization, EuroVis 2014 - State of the Art Reports. Eurographics Association, 2014. URL: https://doi.org/10.2312/eurovisstar.20141174.
  3. Frank Bernhart and Paul C. Kainen. The book thickness of a graph. J. Comb. Theory, Ser. B, 27(3):320-331, 1979. URL: https://doi.org/10.1016/0095-8956(79)90021-2.
  4. Sujoy Bhore, Prosenjit Bose, Pilar Cano, Jean Cardinal, and John Iacono. Dynamic Schnyder woods. CoRR, abs/2106.14451, 2021. URL: http://arxiv.org/abs/2106.14451.
  5. Sujoy Bhore, Robert Ganian, Fabrizio Montecchiani, and Martin Nöllenburg. Parameterized algorithms for book embedding problems. J. Graph Algorithms Appl., 24(4):603-620, 2020. URL: https://doi.org/10.7155/jgaa.00526.
  6. Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, and Hsiang-Yun Wu. Untangling circular drawings: Algorithms and complexity. CoRR, abs/2111.09766, 2021. URL: http://arxiv.org/abs/2111.09766.
  7. Prosenjit Bose, Vida Dujmovic, Ferran Hurtado, Stefan Langerman, Pat Morin, and David R. Wood. A polynomial bound for untangling geometric planar graphs. Discret. Comput. Geom., 42(4):570-585, 2009. URL: https://doi.org/10.1007/s00454-008-9125-3.
  8. Javier Cano, Csaba D. Tóth, and Jorge Urrutia. Upper bound constructions for untangling planar geometric graphs. In Graph Drawing (GD'11), volume 7034 of LNCS, pages 290-295. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-25878-7_28.
  9. Gary Chartrand and Frank Harary. Planar permutation graphs. Annales de l'Institut Henri Poincare. Probabilités et Statistiques, 3:433-438, 1967. Google Scholar
  10. 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. URL: https://doi.org/10.1137/0608002.
  11. Josef Cibulka. Untangling polygons and graphs. Discret. Comput. Geom., 43(2):402-411, 2010. URL: https://doi.org/10.1007/s00454-009-9150-x.
  12. Robert F. Cohen, Giuseppe Di Battista, Roberto Tamassia, and Ioannis G. Tollis. Dynamic graph drawings: Trees, series-parallel digraphs, and planar st-digraphs. SIAM J. Comput., 24(5):970-1001, 1995. URL: https://doi.org/10.1137/S0097539792235724.
  13. Stephan Diehl and Carsten Görg. Graphs, they are changing. In Graph Drawing (GD'02), volume 2528 of LNCS, pages 23-30. Springer, 2002. URL: https://doi.org/10.1007/3-540-36151-0_3.
  14. Mark N. Ellingham, Emily A. Marshall, Kenta Ozeki, and Shoichi Tsuchiya. A characterization of K_2,4-minor-free graphs. SIAM J. Discret. Math., 30(2):955-975, 2016. URL: https://doi.org/10.1137/140986517.
  15. Fabrizio Frati. Planar rectilinear drawings of outerplanar graphs in linear time. In Graph Drawing (GD'20), volume 12590 of LNCS, pages 423-435. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-68766-3_33.
  16. Greg N. Frederickson. Searching among intervals and compact routing tables. Algorithmica, 15(5):448-466, 1996. URL: https://doi.org/10.1007/BF01955044.
  17. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  18. Xavier Goaoc, Jan Kratochvíl, Yoshio Okamoto, Chan-Su Shin, Andreas Spillner, and Alexander Wolff. Untangling a planar graph. Discrete and Computational Geometry, 42(4):542-569, January 2009. URL: https://doi.org/10.1007/s00454-008-9130-6.
  19. Mihyun Kang, Oleg Pikhurko, Alexander Ravsky, Mathias Schacht, and Oleg Verbitsky. Untangling planar graphs from a specified vertex position - hard cases. Discret. Appl. Math., 159(8):789-799, 2011. URL: https://doi.org/10.1016/j.dam.2011.01.011.
  20. Martin Krzywinski, Jacqueline Schein, Inanc Birol, Joseph Connors, Randy Gascoyne, Doug Horsman, Steven J. Jones, and Marco A. Marra. Circos: an information aesthetic for comparative genomics. Genome research, 19(9):1639-1645, 2009. URL: https://doi.org/doi:10.1101/gr.092759.109.
  21. Sylvain Lazard, William J. Lenhart, and Giuseppe Liotta. On the edge-length ratio of outerplanar graphs. Theor. Comput. Sci., 770:88-94, 2019. URL: https://doi.org/10.1016/j.tcs.2018.10.002.
  22. Chun-Cheng Lin, Yi-Yi Lee, and Hsu-Chun Yen. Mental map preserving graph drawing using simulated annealing. Inf. Sci., 181(19):4253-4272, 2011. URL: https://doi.org/10.1016/j.ins.2011.06.005.
  23. Kazuo Misue, Peter Eades, Wei Lai, and Kozo Sugiyama. Layout adjustment and the mental map. J. Visual Languages and Computing, 6(2):183-210, 1995. URL: https://doi.org/10.1006/jvlc.1995.1010.
  24. Andy Nguyen. Solving cyclic longest common subsequence in quadratic time. CoRR, abs/1208.0396, 2012. URL: http://arxiv.org/abs/1208.0396.
  25. János Pach and Gábor Tardos. Untangling a polygon. Discret. Comput. Geom., 28(4):585-592, 2002. URL: https://doi.org/10.1007/s00454-002-2889-y.
  26. Alexander Ravsky and Oleg Verbitsky. On collinear sets in straight-line drawings. In Graph-Theoretic Concepts in Computer Science (WG'11), volume 6986 of LNCS, pages 295-306. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-25870-1_27.
  27. Janet M. Six and Ioannis G. Tollis. A framework and algorithms for circular drawings of graphs. J. Discrete Algorithms, 4(1):25-50, 2006. URL: https://doi.org/10.1016/j.jda.2005.01.009.
  28. Janet M. Six and Ioannis G. Tollis. Circular drawing algorithms. In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization, pages 285-315. Chapman and Hall/CRC, 2013. Google Scholar
  29. Maciej M. Sysło. Characterizations of outerplanar graphs. Discrete Mathematics, 26(1):47-53, 1979. URL: https://doi.org/10.1016/0012-365X(79)90060-8.
  30. Oleg Verbitsky. On the obfuscation complexity of planar graphs. Theor. Comput. Sci., 396(1-3):294-300, 2008. URL: https://doi.org/10.1016/j.tcs.2008.02.032.
  31. Manfred Wiegers. Recognizing outerplanar graphs in linear time. In Graphtheoretic Concepts in Computer Science, International Workshop, WG '86, Germany, 1986, Proceedings, volume 246 of LNCS, pages 165-176. Springer, 1986. URL: https://doi.org/10.1007/3-540-17218-1_57.
  32. Hsiang-Yun Wu, Martin Nöllenburg, and Ivan Viola. Graph models for biological pathway visualization: State of the art and future challenges. In The 1st Workshop on Multilayer Nets: Challenges in Multilayer Network Visualization and Analysis, Vancouver, Canada, October 2019. URL: http://yun-vis.net/projects/bionet/visworkshop2019.pdf.
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