2-Layer Right Angle Crossing Drawings

  • Emilio Di Giacomo
  • Walter Didimo
  • Peter Eades
  • Giuseppe Liotta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)


A 2-layer drawing represents a bipartite graph so that the vertices of each partition set are points of a distinct horizontal line (called a layer) and the edges are straight-line segments. In this paper we study 2-layer drawings where all edge crossings form right angles. We characterize which graphs admit this type of drawing, provide linear-time testing and embedding algorithms, and present a polynomial-time crossing minimization technique. Also, for a given graph G and a constant k, we prove that it is \(\mathcal{NP}\)-complete to decide whether G contains a subgraph of at least k edges having a 2-layer drawing with right angle crossings.


Bipartite Graph Internal Vertex Span Subgraph Edge Crossing Independent Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall (1993)Google Scholar
  2. 2.
    Angelini, P., Cittadini, L., Di Battista, G., Didimo, W., Frati, F., Kaufmann, M., Symvonis, A.: On the perspectives opened by right angle crossing drawings. Journal of Graph Algorithms and Applications 15(1), 53–78 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Argyriou, E.N., Bekos, M.A., Symvonis, A.: The Straight-Line RAC Drawing Problem is NP-Hard. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 74–85. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Didimo, W., Eades, P., Liotta, G.: Drawing Graphs with Right Angle Crossings. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 206–217. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Didimo, W., Eades, P., Liotta, G.: A characterization of complete bipartite RAC graphs. Inf. Process. Lett. 110(16), 687–691 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dujmović, V., Fellows, M.R., Hallett, M.T., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F.A., Suderman, M., Whitesides, S., Wood, D.R.: A fixed-parameter approach to 2-layer planarization. Algorithmica 45(2), 159–182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dujmović, V., Gudmundsson, J., Morin, P., Wolle, T.: Notes on large angle crossing graphs. In: Proceedings of the Sixteenth Symposium on Computing: the Australasian Theory, CATS 2010, vol. 109, pp. 19–24. Australian Computer Society, Inc. (2010)Google Scholar
  8. 8.
    Dujmović, V., Whitesides, S.: An efficient fixed parameter tractable algorithm for 1-sided crossing minimization. Algorithmica 40(1), 15–31 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eades, P., Kelly, D.: Heuristics for drawing 2-layered networks. Ars Comb. 21, 89–98 (1986)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Eades, P., Liotta, G.: Right angle crossing graphs and 1-planarity. In: EuroCG (2011)Google Scholar
  11. 11.
    Eades, P., McKay, B., Wormald, N.: On an edge crossing problem. In: Proc. of 9th Australian Computer Science Conference, pp. 327–334 (1986)Google Scholar
  12. 12.
    Eades, P., Whitesides, S.: Drawing graphs in two layers. Theoretical Computer Science 131(2), 361–374 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11(4), 379–403 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Harary, F., Schwenk, A.: A new crossing number for bipartite graphs. Utilitas Mathematica 1, 203–209 (1972)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huang, W.: Using eye tracking to investigate graph layout effects. In: APVIS, pp. 97–100 (2007)Google Scholar
  16. 16.
    Huang, W., Hong, S.-H., Eades, P.: Effects of crossing angles. In: PacificVis, pp. 41–46 (2008)Google Scholar
  17. 17.
    Jünger, M., Mutzel, P.: 2-layer straightline crossing minimization: Performance of exact and heuristic algorithms. J. Graph Algorithms Appl. 1 (1997)Google Scholar
  18. 18.
    Mutzel, P.: An alternative method to crossing minimization on hierarchical graphs. SIAM J. on Optimization 11(4), 1065–1080 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tomii, N., Kambayashi, Y., Yajima, S.: On planarization algorithms of 2-level graphs. Technical Report EC77-38, Inst. of Elect. and Comm. Eng. Japan (1977)Google Scholar
  20. 20.
    Valls, V., Martí, R., Lino, P.: A branch and bound algorithm for minimizing the number of crossing arcs in bipartite graphs. Europ. J. of Oper. Res. 90(2), 303–319 (1996)CrossRefzbMATHGoogle Scholar
  21. 21.
    van Kreveld, M.: The Quality Ratio of RAC Drawings and Planar Drawings of Planar Graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 371–376. Springer, Heidelberg (2011)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Walter Didimo
    • 1
  • Peter Eades
    • 2
  • Giuseppe Liotta
    • 1
  1. 1.Università di PerugiaItaly
  2. 2.University of SydneyAustralia

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