Document Open Access Logo

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Authors Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, Siddharth Gupta

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2019.9.pdf
  • Filesize: 1.48 MB
  • 17 pages

Document Identifiers

Author Details

Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
David Eppstein
  • University of California, Irvine, USA
Michael T. Goodrich
  • University of California, Irvine, USA
Siddharth Gupta
  • Ben-Gurion University of the Negev, Beersheba, Israel

Funding

  • Da Lozzo, Giordano: Supported in part by H2020-MSCA-RISE project 734922 – "CONNECT", by MIUR Project "AHeAD" under PRIN 20174LF3T8, by MIUR Project "MODE" under PRIN 20157EFM5C, and by MIUR-DAAD JMP N^o 34120.
  • Eppstein, David: Supported in part by the US NSF under grants CCF-1618301 and CCF-1616248.
  • Goodrich, Michael T.: Supported in part by the US NSF under grant 1815073.
  • Gupta, Siddharth: Supported in part by the Zuckerman STEM Leadership Program and by the Frankel Foundation.

Cite AsGet BibTex

Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, and Siddharth Gupta. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2019.9

Abstract

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA'95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph theory
Keywords
  • Clustered planarity
  • carving-width
  • non-crossing partitions
  • FPT

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Isolde Adler, Binh-Minh Bui-Xuan, Yuri Rabinovich, Gabriel Renault, Jan Arne Telle, and Martin Vatshelle. On the Boolean-Width of a Graph: Structure and Applications. In Dimitrios M. Thilikos, editor, WG 2010, volume 6410 of LNCS, pages 159-170, 2010. URL: https://doi.org/10.1007/978-3-642-16926-7_16.
  2. Hugo A. Akitaya, Radoslav Fulek, and Csaba D. Tóth. Recognizing Weak Embeddings of Graphs. In Artur Czumaj, editor, SODA '18, pages 274-292. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.20.
  3. Patrizio Angelini and Giordano Da Lozzo. SEFE = C-Planarity? Comput. J., 59(12):1831-1838, 2016. URL: https://doi.org/10.1093/comjnl/bxw035.
  4. Patrizio Angelini and Giordano Da Lozzo. Clustered Planarity with Pipes. Algorithmica, 81(6):2484-2526, 2019. URL: https://doi.org/10.1007/s00453-018-00541-w.
  5. 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: https://doi.org/10.1007/s00453-016-0128-9.
  6. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. Relaxing the constraints of clustered planarity. Comput. Geom., 48(2):42-75, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.08.001.
  7. 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: https://doi.org/10.7155/jgaa.00437.
  8. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, and Vincenzo Roselli. The importance of being proper: (In clustered-level planarity and T-level planarity). Theor. Comput. Sci., 571:1-9, 2015. URL: https://doi.org/10.1016/j.tcs.2014.12.019.
  9. Patrizio Angelini, Fabrizio Frati, and Michael Kaufmann. Straight-Line Rectangular Drawings of Clustered Graphs. Discrete & Computational Geometry, 45(1):88-140, 2011. URL: https://doi.org/10.1007/s00454-010-9302-z.
  10. Jan Christoph Athenstädt and Sabine Cornelsen. Planarity of Overlapping Clusterings Including Unions of Two Partitions. J. Graph Algorithms Appl., 21(6):1057-1089, 2017. URL: https://doi.org/10.7155/jgaa.00450.
  11. T. Biedl. Drawing Planar Partitions III: Two Constrained Embedding Problems. Tech. Report RRR 13-98, Rutcor Research Report, 1998. Google Scholar
  12. Therese C. Biedl and Martin Vatshelle. The Point-Set Embeddability Problem for Plane graphs. Int. J. Comput. Geometry Appl., 23(4-5):357-396, 2013. URL: https://doi.org/10.1142/S0218195913600091.
  13. Thomas Bläsius and Ignaz Rutter. A new perspective on clustered planarity as a combinatorial embedding problem. Theor. Comput. Sci., 609:306-315, 2016. URL: https://doi.org/10.1016/j.tcs.2015.10.011.
  14. Glencora Borradaile, Jeff Erickson, Hung Le, and Robbie Weber. Embedded-width: A variant of treewidth for plane graphs, 2017. URL: http://arxiv.org/abs/1703.07532.
  15. Vincent Bouchitté, Frédéric Mazoit, and Ioan Todinca. Treewidth of planar graphs: connections with duality. ENDM, 10:34-38, 2001. URL: https://doi.org/10.1016/S1571-0653(04)00353-1.
  16. Franz-Josef Brandenburg, David Eppstein, Michael T. Goodrich, Stephen G. Kobourov, Giuseppe Liotta, and Petra Mutzel. Selected Open Problems in Graph Drawing. In Giuseppe Liotta, editor, GD '03, volume 2912 of LNCS, pages 515-539. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-24595-7_55.
  17. Markus Chimani, Giuseppe Di Battista, Fabrizio Frati, and Karsten Klein. Advances on Testing C-Planarity of Embedded Flat Clustered Graphs. In Christian A. Duncan and Antonios Symvonis, editors, GD '14, volume 8871 of LNCS, pages 416-427. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-45803-7_35.
  18. Markus Chimani and Karsten Klein. Shrinking the Search Space for Clustered Planarity. In Walter Didimo and Maurizio Patrignani, editors, GD '12, volume 7704 of LNCS, pages 90-101. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-36763-2_9.
  19. Derek G. Corneil and Udi Rotics. On the Relationship Between Clique-Width and Treewidth. SIAM J. Comput., 34(4):825-847, 2005. URL: https://doi.org/10.1137/S0097539701385351.
  20. Sabine Cornelsen and Dorothea Wagner. Completely connected clustered graphs. J. Discrete Algorithms, 4(2):313-323, 2006. URL: https://doi.org/10.1016/j.jda.2005.06.002.
  21. Pier Francesco Cortese and Giuseppe Di Battista. Clustered planarity. In Joseph S. B. Mitchell and Günter Rote, editors, SoCG '05, pages 32-34. ACM, 2005. URL: https://doi.org/10.1145/1064092.1064093.
  22. Pier Francesco Cortese, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Maurizio Pizzonia. C-Planarity of C-Connected Clustered Graphs. J. Graph Algorithms Appl., 12(2):225-262, 2008. URL: http://jgaa.info/accepted/2008/Cortese+2008.12.2.pdf, URL: https://doi.org/10.7155/jgaa.00165.
  23. Pier Francesco Cortese, Giuseppe Di Battista, Maurizio Patrignani, and Maurizio Pizzonia. On embedding a cycle in a plane graph. Discrete Mathematics, 309(7):1856-1869, 2009. URL: https://doi.org/10.1016/j.disc.2007.12.090.
  24. Pier Francesco Cortese and Maurizio Patrignani. Clustered Planarity = Flat Clustered Planarity. In Therese C. Biedl and Andreas Kerren, editors, GD 2018, volume 11282 of LNCS, pages 23-38. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-04414-5_2.
  25. Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignani. Computing NodeTrix Representations of Clustered Graphs. J. Graph Algorithms Appl., 22(2):139-176, 2018. URL: https://doi.org/10.7155/jgaa.00461.
  26. Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, and Siddharth Gupta. Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity. In Andreas Brandstädt, Ekkehard Köhler, and Klaus Meer, editors, WG 2018, volume 11159 of LNCS, pages 111-124. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-00256-5_10.
  27. Elias Dahlhaus. A Linear Time Algorithm to Recognize Clustered Graphs and Its Parallelization. In Claudio L. Lucchesi and Arnaldo V. Moura, editors, LATIN '98, volume 1380 of LNCS, pages 239-248. Springer, 1998. URL: https://doi.org/10.1007/BFb0054325.
  28. Giuseppe Di Battista, Walter Didimo, and A. Marcandalli. Planarization of Clustered Graphs. In Petra Mutzel, Michael Jünger, and Sebastian Leipert, editors, GD '01, volume 2265 of LNCS, pages 60-74. Springer, 2001. URL: https://doi.org/10.1007/3-540-45848-4_5.
  29. Giuseppe Di Battista and Fabrizio Frati. Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces. J. Graph Algorithms Appl., 13(3):349-378, 2009. URL: http://jgaa.info/accepted/2009/DiBattistaFrati2009.13.3.pdf, URL: https://doi.org/10.7155/jgaa.00191.
  30. Walter Didimo, Francesco Giordano, and Giuseppe Liotta. Overlapping Cluster Planarity. J. Graph Algorithms Appl., 12(3):267-291, 2008. URL: http://jgaa.info/accepted/2008/DidimoGiordanoLiotta2008.12.3.pdf, URL: https://doi.org/10.7155/jgaa.00167.
  31. Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for Clustered Graphs. In Paul G. Spirakis, editor, ESA'95, volume 979 of LNCS, pages 213-226. Springer, 1995. URL: https://doi.org/10.1007/3-540-60313-1_145.
  32. Michael Forster and Christian Bachmaier. Clustered Level Planarity. In Peter van Emde Boas, Jaroslav Pokorný, Mária Bieliková, and Julius Stuller, editors, SOFSEM '04, volume 2932 of LNCS, pages 218-228. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24618-3_18.
  33. Radoslav Fulek and Jan Kyncl. Hanani-Tutte for Approximating Maps of Graphs. In Bettina Speckmann and Csaba D. Tóth, editors, SoCG '18, volume 99 of LIPIcs, pages 39:1-39:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.39.
  34. Radoslav Fulek, Jan Kyncl, Igor Malinovic, and Dömötör Pálvölgyi. Efficient c-planarity testing algebraically. CoRR, abs/1305.4519, 2013. URL: http://arxiv.org/abs/1305.4519.
  35. Radoslav Fulek, Jan Kyncl, Igor Malinovic, and Dömötör Pálvölgyi. Clustered Planarity Testing Revisited. Electr. J. Comb., 22(4):P4.24, 2015. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p24.
  36. Radoslav Fulek and Csaba D. Tóth. Atomic Embeddability, Clustered Planarity, and Thickenability. CoRR, abs/1907.13086, 2019. URL: http://arxiv.org/abs/1907.13086.
  37. Christopher D. Godsil and Gordon F. Royle. Algebraic Graph Theory. Graduate texts in mathematics. Springer, 2001. URL: https://doi.org/10.1007/978-1-4613-0163-9.
  38. Michael T. Goodrich, George S. Lueker, and Jonathan Z. Sun. C-Planarity of Extrovert Clustered Graphs. In Patrick Healy and Nikola S. Nikolov, editors, GD '05, volume 3843 of LNCS, pages 211-222. Springer, 2005. URL: https://doi.org/10.1007/11618058_20.
  39. Qian-Ping Gu and Hisao Tamaki. Optimal branch-decomposition of planar graphs in O(n³) Time. ACM Trans. Algorithms, 4(3):30:1-30:13, 2008. URL: https://doi.org/10.1145/1367064.1367070.
  40. Carsten Gutwenger, Michael Jünger, Sebastian Leipert, Petra Mutzel, Merijam Percan, and René Weiskircher. Advances in C-Planarity Testing of Clustered Graphs. In Stephen G. Kobourov and Michael T. Goodrich, editors, GD '02, volume 2528 of LNCS, pages 220-235. Springer, 2002. URL: https://doi.org/10.1007/3-540-36151-0_21.
  41. Carsten Gutwenger, Petra Mutzel, and Marcus Schaefer. Practical Experience with Hanani-Tutte for Testing c-Planarity. In Catherine C. McGeoch and Ulrich Meyer, editors, ALENEX '14, pages 86-97. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973198.9.
  42. Seok-Hee Hong and Hiroshi Nagamochi. Convex drawings of hierarchical planar graphs and clustered planar graphs. J. Discrete Algorithms, 8(3):282-295, 2010. URL: https://doi.org/10.1016/j.jda.2009.05.003.
  43. Seok-Hee Hong and Hiroshi Nagamochi. Simpler algorithms for testing two-page book embedding of partitioned graphs. Theoretical Computer Science, 2016. URL: https://doi.org/10.1016/j.tcs.2015.12.039.
  44. John E. Hopcroft and Robert Endre Tarjan. Efficient Algorithms for Graph Manipulation [H] (Algorithm 447). Commun. ACM, 16(6):372-378, 1973. URL: https://doi.org/10.1145/362248.362272.
  45. Vít Jelínek, Eva Jelínková, Jan Kratochvíl, and Bernard Lidický. Clustered Planarity: Embedded Clustered Graphs with Two-Component Clusters. In Ioannis G. Tollis and Maurizio Patrignani, editors, GD '08, volume 5417 of LNCS, pages 121-132. Springer, 2008. URL: https://doi.org/10.1007/978-3-642-00219-9_13.
  46. Eva Jelínková, Jan Kára, Jan Kratochvíl, Martin Pergel, Ondrej Suchý, and Tomás Vyskocil. Clustered Planarity: Small Clusters in Cycles and Eulerian Graphs. J. Graph Algorithms Appl., 13(3):379-422, 2009. URL: http://jgaa.info/accepted/2009/Jelinkova+2009.13.3.pdf, URL: https://doi.org/10.7155/jgaa.00192.
  47. Giordano Da Lozzo, David Eppstein, Michael T. Goodrich, and Siddharth Gupta. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width, 2019. URL: http://arxiv.org/abs/1910.02057.
  48. Hiroshi Nagamochi and Katsutoshi Kuroya. Drawing c-planar biconnected clustered graphs. Discrete Applied Mathematics, 155(9):1155-1174, 2007. URL: https://doi.org/10.1016/j.dam.2006.04.044.
  49. Sang-il Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory, 57(3):239-244, 2008. URL: https://doi.org/10.1002/jgt.20280.
  50. 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.
  51. Juanjo Rué, Ignasi Sau, and Dimitrios M. Thilikos. Dynamic programming for graphs on surfaces. ACM Trans. Algorithms, 10(2):8:1-8:26, 2014. URL: https://doi.org/10.1145/2556952.
  52. Róbert Sasák. Comparing 17 graph parameters. Master’s thesis, Department of Informatics, University of Bergen, Bergen, Norway, 2010. Google Scholar
  53. Marcus Schaefer. Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants. J. Graph Algorithms Appl., 17(4):367-440, 2013. URL: https://doi.org/10.7155/jgaa.00298.
  54. Paul D. Seymour and Robin Thomas. Call Routing and the Ratcatcher. Combinatorica, 14(2):217-241, 1994. URL: https://doi.org/10.1007/BF01215352.
  55. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Constructive Linear Time Algorithms for Small Cutwidth and Carving-Width. In D. T. Lee and Shang-Hua Teng, editors, ISAAC '00, volume 1969 of LNCS, pages 192-203. Springer, 2000. URL: https://doi.org/10.1007/3-540-40996-3_17.
  56. Juan José Besa Vial, Giordano Da Lozzo, and Michael T. Goodrich. Computing k-Modal Embeddings of Planar Digraphs. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, ESA 2019, volume 144 of LIPIcs, pages 17:1-17:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.17.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact