Simultaneous Orthogonal Planarity

  • Patrizio Angelini
  • Steven Chaplick
  • Sabine Cornelsen
  • Giordano Da Lozzo
  • Giuseppe Di Battista
  • Peter Eades
  • Philipp Kindermann
  • Jan Kratochvíl
  • Fabian Lipp
  • Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

We introduce and study the \({\textsc {OrthoSEFE}\text {-}{k}} \) problem: Given k planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the k graphs? We show that the problem is NP-complete for \(k \ge 3\) even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for \(k \ge 2\) even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for \(k=2\) when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.

References

  1. 1.
    Angelini, P., Chaplick, S., Cornelsen, S., Da Lozzo, G., Di Battista, G., Eades, P., Kindermann, P., Kratochvíl, J., Lipp, F.: Simultaneous Orthogonal Planarity. ArXiv e-prints, abs/1608.08427 (2016)Google Scholar
  2. 2.
    Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Beyond level planarity. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 482–495. Springer, Heidelberg (2016)Google Scholar
  3. 3.
    Angelini, P., Da Lozzo, G., Neuwirth, D.: Advancements on SEFE and partitioned book embedding problems. Theoret. Comput. Sci. 575, 71–89 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Angelini, P., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph. J. Discrete Algorithms 14, 150–172 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Argyriou, E.N., Bekos, M.A., Kaufmann, M., Symvonis, A.: Geometric RAC simultaneous drawings of graphs. J. Graph Algorithms Appl. 17(1), 11–34 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Auslander, L., Parter, S.V.: On embedding graphs in the sphere. J. Math. Mech. 10(3), 517–523 (1961)MathSciNetMATHGoogle Scholar
  7. 7.
    Bekos, M.A., van Dijk, T.C., Kindermann, P., Wolff, A.: Simultaneous drawing of planar graphs with right-angle crossings and few bends. J. Graph Algorithms Appl. 20(1), 133–158 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Biedl, T., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bläsius, T., Karrer, A., Rutter, I.: Simultaneous embedding: edge orderings, relative positions, cutvertices. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 220–231. Springer, Heidelberg (2013). doi:10.1007/978-3-319-03841-4_20 CrossRefGoogle Scholar
  10. 10.
    Bläsius, T., Karrer, A., Rutter, I.: Simultaneous embedding: Edge orderings, relative positions, cutvertices. ArXiv e-prints, abs/1506.05715 (2015)Google Scholar
  11. 11.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R., (ed.) Handbook of Graph Drawing and Visualization. CRC Press (2013)Google Scholar
  12. 12.
    Bläsius, T., Rutter, I.: Disconnectivity and relative positions in simultaneous embeddings. Comput. Geom. 48(6), 459–478 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. ACM Trans. Algorithms 12(2), 16 (2016)MathSciNetGoogle Scholar
  14. 14.
    Brandes, U.: Eager st-ordering. In: Möhring, R., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 247–256. Springer, Heidelberg (2002). doi:10.1007/3-540-45749-6_25 CrossRefGoogle Scholar
  15. 15.
    Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Estrella-Balderrama, A., Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M.: Simultaneous geometric graph embeddings. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 280–290. Springer, Heidelberg (2008). doi:10.1007/978-3-540-77537-9_28 CrossRefGoogle Scholar
  18. 18.
    Haeupler, B., Jampani, K.R., Lubiw, A.: Testing simultaneous planarity when the common graph is 2-connected. J. Graph Algorithms Appl. 17(3), 147–171 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jampani, K.R., Lubiw, A.: Simultaneous interval graphs. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 206–217. Springer, Heidelberg (2010). doi:10.1007/978-3-642-17517-6_20 CrossRefGoogle Scholar
  20. 20.
    Jampani, K.R., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. J. Graph Algorithms Appl. 16(2), 283–315 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jünger, M., Schulz, M.: Intersection graphs in simultaneous embedding with fixed edges. J. Graph Algorithms Appl. 13(2), 205–218 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Moret, B.M.E.: Planar NAE3SAT is in P. ACM SIGACT News 19(2), 51–54 (1988)CrossRefMATHGoogle Scholar
  23. 23.
    Moret, B.M.E.: Theory of Computation. Addison-Wesley-Longman, Reading (1998)MATHGoogle Scholar
  24. 24.
    Papadimitriou, C.H.: Computational Complexity. Academic Internet Publ., London (2007)MATHGoogle Scholar
  25. 25.
    Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algorithms Appl. 17(4), 367–440 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.), STOC 1978, pp. 216–226. ACM (1978)Google Scholar
  27. 27.
    Shih, W., Wu, S., Kuo, Y.: Unifying maximum cut and minimum cut of a planar graph. IEEE Trans. Comput. 39(5), 694–697 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Steven Chaplick
    • 2
  • Sabine Cornelsen
    • 3
  • Giordano Da Lozzo
    • 4
  • Giuseppe Di Battista
    • 4
  • Peter Eades
    • 5
  • Philipp Kindermann
    • 6
  • Jan Kratochvíl
    • 7
  • Fabian Lipp
    • 2
  • Ignaz Rutter
    • 8
  1. 1.Universität TübingenTübingenGermany
  2. 2.Universität WürzburgWürzburgGermany
  3. 3.Universität KonstanzKonstanzGermany
  4. 4.Roma Tre UniversityRomeItaly
  5. 5.The University of SydneySydneyAustralia
  6. 6.FernUniversität in HagenHagenGermany
  7. 7.Charles UniversityPragueCzech Republic
  8. 8.Karlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations