Advertisement

Drawing Non-Planar Graphs with Crossing-Free Subgraphs

  • Patrizio Angelini
  • Carla Binucci
  • Giordano Da Lozzo
  • Walter Didimo
  • Luca Grilli
  • Fabrizio Montecchiani
  • Maurizio Patrignani
  • Ioannis G. Tollis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

Abstract

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ? We give positive and negative results for different kinds of spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G ∖ S; in this setting different trade-offs between number of bends and drawing area are given.

Keywords

Span Tree Planar Graph Geometric Graph Span Subgraph Graph Drawing 
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.

References

  1. 1.
    Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete & Computational Geometry 41(3), 365–375 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. Journal of Combinatorial Theory, Ser. A 114(3), 563–571 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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. Journal of Discrete Algorithms 14, 150–172 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Angelini, P., Binucci, C., Da Lozzo, G., Didimo, W., Grilli, L., Montecchiani, F., Patrignani, M., Tollis, I.G.: Drawings of non-planar graphs with crossing-free subgraphs. ArXiv e-prints 1308.6706 (September 2013)Google Scholar
  5. 5.
    Bárány, I., Rote, G.: Strictly convex drawings of planar graphs. Documenta. Math. 11, 369–391 (2006)zbMATHGoogle Scholar
  6. 6.
    Blasiüs, 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
  7. 7.
    Brandenburg, F.J., Eppstein, D., Gleißner, A., Goodrich, M.T., Hanauer, K., Reislhuber, J.: On the density of maximal 1-planar graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 327–338. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Buchheim, C., Chimani, M., Gutwenger, C., Jünger, M., Mutzel, P.: Crossings and planarization. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. CRC Press (2013)Google Scholar
  9. 9.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)zbMATHGoogle Scholar
  10. 10.
    Di Giacomo, E., Didimo, W., Liotta, G., Montecchiani, F.: h-quasi planar drawings of bounded treewidth graphs in linear area. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 91–102. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Di Giacomo, E., Didimo, W., Liotta, G., Montecchiani, F.: Area requirement of graph drawings with few crossings per edge. Computational Geometry 46(8), 909–916 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Didimo, W.: Density of straight-line 1-planar graph drawings. Information Processing Letters 113(7), 236–240 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theoretical Computer Science 412(39), 5156–5166 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Didimo, W., Liotta, G.: The crossing angle resolution in graph drawing. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory. Springer (2013)Google Scholar
  15. 15.
    Eades, P., Hong, S.H., Katoh, N., Liotta, G., Schweitzer, P., Suzuki, Y.: Testing maximal 1-planarity of graphs with a rotation system in linear time - (extended abstract). In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 339–345. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    Hong, S.-H., Eades, P., Liotta, G., Poon, S.-H.: Fáry’s theorem for 1-planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Jansen, K., Woeginger, G.J.: The complexity of detecting crossingfree configurations in the plane. BIT Numerical Mathematics 33(4), 580–595 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Knauer, C., Schramm, É., Spillner, A., Wolff, A.: Configurations with few crossings in topological graphs. Computational Geometry 37(2), 104–114 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. Journal of Graph Theory 72(1), 30–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kowalik, L., Kurowski, M.: Short path queries in planar graphs in constant time. In: Larmore, L.L., Goemans, M.X. (eds.) STOC 2003, pp. 143–148. ACM (2003)Google Scholar
  22. 22.
    Kratochvìl, J., Lubiv, A., Nešetřil, J.: Noncrossing subgraphs in topological layouts. SIAM Journal on Discrete Mathematics 4(2), 223–244 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. Algorithmica 16(1), 111–117 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rivera-Campo, E., Urrutia-Galicia, V.: A sufficient condition for the existence of plane spanning trees on geometric graphs. Computational Geometry 46(1), 1–6 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Suk, A.: k-quasi-planar graphs. In: Speckmann, B. (ed.) GD 2011. LNCS, vol. 7034, pp. 266–277. Springer, Heidelberg (2011)Google Scholar
  27. 27.
    Tutte, W.T.: How to draw a graph. Proceedings of the London Mathematical Society s3-13(1), 743–767 (1963)Google Scholar
  28. 28.
    Valtr, P.: On geometric graphs with no k pairwise parallel edges. Discrete & Computational Geometry 19(3), 461–469 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Carla Binucci
    • 2
  • Giordano Da Lozzo
    • 1
  • Walter Didimo
    • 2
  • Luca Grilli
    • 2
  • Fabrizio Montecchiani
    • 2
  • Maurizio Patrignani
    • 1
  • Ioannis G. Tollis
    • 3
  1. 1.Università Roma TreItaly
  2. 2.Università degli Studi di PerugiaItaly
  3. 3.Institute of Computer Science-FORTHUniv. of CreteGreece

Personalised recommendations