Computing k-Modal Embeddings of Planar Digraphs

Authors Juan José Besa , Giordano Da Lozzo , Michael T. Goodrich

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Author Details

Juan José Besa
  • University of California, Irvine, USA
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Michael T. Goodrich
  • University of California, Irvine, USA

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Juan José Besa, Giordano Da Lozzo, and Michael T. Goodrich. Computing k-Modal Embeddings of Planar Digraphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks. First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. []. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Mathematics of computing → Graph algorithms
  • Information systems → Data structures
  • Modal Embeddings
  • Planarity
  • Directed Graphs
  • SPQR trees


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  1. Patrizio Angelini, Giordano Da Lozzo, Marco Di Bartolomeo, Valentino Di Donato, Maurizio Patrignani, Vincenzo Roselli, and Ioannis G. Tollis. Algorithms and Bounds for L-Drawings of Directed Graphs. Int. J. of Foundations of Computer Science, 29(04):461-480, 2018. URL:
  2. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. Beyond Level Planarity. In Yifan Hu and Martin Nöllenburg, editors, GD '16, volume 9801 of LNCS, pages 482-495. Springer, 2016. URL:
  3. 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:
  4. 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. Google Scholar
  5. Patrizio Angelini, Peter Eades, Seok-Hee Hong, Karsten Klein, Stephen G. Kobourov, Giuseppe Liotta, Alfredo Navarra, and Alessandra Tappini. Turning Cliques into Paths to Achieve Planarity. In Therese C. Biedl and Andreas Kerren, editors, GD 2018, volume 11282 of LNCS, pages 67-74. Springer, 2018. URL:
  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, September 2018. URL:
  7. Christian Bachmaier, Franz-Josef Brandenburg, and Michael Forster. Radial Level Planarity Testing and Embedding in Linear Time. J. Graph Algorithms Appl., 9(1):53-97, 2005. URL:, URL:
  8. Paola Bertolazzi, Giuseppe Di Battista, Giuseppe Liotta, and Carlo Mannino. Upward Drawings of Triconnected Digraphs. Algorithmica, 12(6):476-497, 1994. URL:
  9. Carla Binucci, Walter Didimo, and Francesco Giordano. Maximum upward planar subgraphs of embedded planar digraphs. Comput. Geom., 41(3):230-246, 2008. Google Scholar
  10. Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward planarity testing of embedded mixed graphs. Theor. Comput. Sci., 526:75-89, 2014. URL:
  11. Kellogg S. Booth and George S. Lueker. Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms. J. Comput. Syst. Sci., 13(3):335-379, 1976. URL:
  12. Guido Brückner and Ignaz Rutter. Partial and Constrained Level Planarity. In Philip N. Klein, editor, SODA '17, pages 2000-2011. SIAM, 2017. URL:
  13. 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 '17, volume 10692 of LNCS, pages 465-478. Springer, 2017. URL:
  14. 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. Google Scholar
  15. Giuseppe Di Battista and Enrico Nardelli. Hierarchies and planarity theory. IEEE Trans. Systems, Man, and Cybernetics, 18(6):1035-1046, 1988. URL:
  16. Giuseppe Di Battista and Roberto Tamassia. On-Line Graph Algorithms with SPQR-Trees. In Mike Paterson, editor, ICALP '90, volume 443 of LNCS, pages 598-611. Springer, 1990. URL:
  17. Ashim Garg and Roberto Tamassia. On the Computational Complexity of Upward and Rectilinear Planarity Testing. SIAM J. Comput., 31(2):601-625, 2001. URL:
  18. Emilio Di Giacomo, Giuseppe Liotta, Maurizio Patrignani, and Alessandra Tappini. NodeTrix Planarity Testing with Small Clusters. In Fabrizio Frati and Kwan-Liu Ma, editors, GD '17, volume 10692 of LNCS, pages 479-491. Springer, 2017. URL:
  19. Nathalie Henry, Jean-Daniel Fekete, and Michael J. McGuffin. NodeTrix: a Hybrid Visualization of Social Networks. IEEE Trans. Vis. Comput. Graph., 13(6):1302-1309, 2007. URL:
  20. John E. Hopcroft and Robert Endre Tarjan. Efficient Planarity Testing. J. ACM, 21(4):549-568, 1974. URL:
  21. Juan José Besa, Giordano Da Lozzo, and Michael T. Goodrich. Computing k-Modal Embeddings of Planar Digraphs. CoRR, abs/1907.01630, 2019. URL:
  22. Michael Jünger, Sebastian Leipert, and Petra Mutzel. Level Planarity Testing in Linear Time. In Sue Whitesides, editor, GD '98, volume 1547 of LNCS, pages 224-237. Springer, 1998. URL:
  23. Boris Klemz and Günter Rote. Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity. In Fabrizio Frati and Kwan-Liu Ma, editors, GD '17, volume 10692 of LNCS, pages 440-453. Springer, 2017. URL:
  24. JiaWei Lu, Yuanming Zhang, Jun Xu, Gang Xiao, and Qianhui Althea Liang. Data Visualization of Web Service with Parallel Coordinates and NodeTrix. In IEEE International Conference on Services Computing, SCC 2014, Anchorage, AK, USA, June 27 - July 2, 2014, pages 766-773. IEEE Computer Society, 2014. URL:
  25. B. M. E. Moret. Planar NAE3SAT is in P. SIGACT News, 19(2):51-54, June 1988. URL:
  26. Stefan Porschen, Bert Randerath, and Ewald Speckenmeyer. Linear Time Algorithms for Some Not-All-Equal Satisfiability Problems. In Enrico Giunchiglia and Armando Tacchella, editors, SAT '03, volume 2919 of LNCS, pages 172-187. Springer, 2003. URL:
  27. Xinsong Yang, Lei Shi, Madelaine Daianu, Hanghang Tong, Qingsong Liu, and Paul M. Thompson. Blockwise Human Brain Network Visual Comparison Using NodeTrix Representation. IEEE Trans. Vis. Comput. Graph., 23(1):181-190, 2017. URL:
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