International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 166-179 | Cite as

Simultaneous Embeddings with Few Bends and Crossings

  • Fabrizio Frati
  • Michael Hoffmann
  • Vincent Kusters
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

A simultaneous embedding with fixed edges (Sefe) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to their common vertices and edges. We show that whenever R and B admit a Sefe, they also admit a Sefe in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if R and B are trees then one bend per edge and four crossings per edge pair suffice, (2) if R is a planar graph and B is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if R and B are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. This improves on results by Grilli et al. (GD’14), who prove that nine bends per edge suffice, and by Chan et al. (GD’14), who prove that twenty-four crossings per edge pair suffice.

Notes

Acknowledgments

This research initiated at the Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Barbados in March 2015. The authors thank the other participants for a stimulating atmosphere. Frati also wishes to thank Anna Lubiw and Marcus Schaefer for insightful ideas they shared during the research for [7].

References

  1. 1.
    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. Discr. Algorithms 14, 150–172 (2012)CrossRefMATHGoogle Scholar
  2. 2.
    Angelini, P., Geyer, M., Kaufmann, M., Neuwirth, D.: On a tree and a path with no geometric simultaneous embedding. J. Graph Algorithms Appl. 16(1), 37–83 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bekos, M.A., van Dijk, T.C., Kindermann, P., Wolff, A.: Simultaneous drawing of planar graphs with right-angle crossings and few bends. In: Rahman, M.S., Tomita, E. (eds.) WALCOM 2015. LNCS, vol. 8973, pp. 222–233. Springer, Heidelberg (2015) Google Scholar
  4. 4.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embeddings of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, Discrete Mathematics and Its Applications, chapter 11, pp. 349–382. Chapman and Hall/CRC (2013)Google Scholar
  5. 5.
    Bläsius, T., Rutter, I.: Disconnectivity and relative positions in simultaneous embeddings. Comp. Geom. 48(6), 459–478 (2015)CrossRefGoogle Scholar
  6. 6.
    Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D.P., Kobourov, S.G., Lubiw, A., Mitchell, J.S.: On simultaneous planar graph embeddings. Comput. Geom. Theory Appl. 36(2), 117–130 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chan, T.M., Frati, F., Gutwenger, C., Lubiw, A., Mutzel, P., Schaefer, M.: Drawing partially embedded and simultaneously planar graphs. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 25–39. Springer, Heidelberg (2014) Google Scholar
  8. 8.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl. 30(1), 1–23 (2005)CrossRefMATHGoogle Scholar
  9. 9.
    Di Giacomo, E., Liotta, G.: Simultaneous embedding of outerplanar graphs, paths, and cycles. Int. J. Comput. Geom. Appl. 17(2), 139–160 (2007)CrossRefMATHGoogle Scholar
  10. 10.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. Graph Algorithms Appl. 9(3), 347–364 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Erten, C., Kobourov, S.G., Le, V., Navabi, A.: Simultaneous graph drawing: layout algorithms and visualization schemes. J. Graph Algorithms Appl. 9(1), 165–182 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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) CrossRefGoogle Scholar
  13. 13.
    Frati, F.: Embedding graphs simultaneously with fixed edges. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 108–113. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  14. 14.
    Frati, F., Hoffmann, M., Kusters, V.: Simultaneous embeddings with few bends and crossings (2015). CoRR abs/1508.07921
  15. 15.
    Frati, F., Kaufmann, M., Kobourov, S.G.: Constrained simultaneous and near-simultaneous embeddings. J. Graph Algorithms Appl. 13(3), 447–465 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Geyer, M., Kaufmann, M., Vrt’o, I.: Two trees which are self-intersecting when drawn simultaneously. Discrete Math. 309(7), 1909–1916 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Grilli, L., Hong, S.-H., Kratochvíl, J., Rutter, I.: Drawing simultaneously embedded graphs with few bends. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 40–51. Springer, Heidelberg (2014) Google Scholar
  18. 18.
    Hong, S., Nagamochi, H.: An algorithm for constructing star-shaped drawings of plane graphs. Comput. Geom. Theory Appl. 43(2), 191–206 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kammer, F.: Simultaneous embedding with two bends per edge in polynomial area. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 255–267. Springer, Heidelberg (2006) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Fabrizio Frati
    • 1
  • Michael Hoffmann
    • 2
  • Vincent Kusters
    • 2
  1. 1.Dipartimento di IngegneriaUniversity Roma TreRomeItaly
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland

Personalised recommendations