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International Symposium on Graph Drawing

GD 2014: Graph Drawing pp 186–197Cite as

Fan-Planar Graphs: Combinatorial Properties and Complexity Results

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8871)

Abstract

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every n-vertex fan-planar drawing has at most 5n − 10 edges, and that this bound is tight for n ≥ 20. We extend their result from both the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and k-planarity. Also, we prove that testing fan-planarity in the variable embedding setting is NP-complete.

Keywords

  • Complexity Result
  • Combinatorial Property
  • Outer Face
  • Geometric Graph
  • Outerplanar Graph

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.

Research supported in part by the MIUR project AMANDA “Algorithmics for MAssive and Networked DAta”, prot. 2012C4E3KT_001. This work started at the Bertinoro Workshop on Graph Drawing 2014. We thank Michael Kaufmann and Torsten Ueckerdt for suggesting the study of fan-planar graphs during the workshop. We also thank all the participants of the workshop for the useful discussions on this topic.

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References

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Ackerman, E., Fulek, R., Tóth, C.D.: Graphs that admit polyline drawings with few crossing angles. SIAM J. on Discrete Mathematics 26(1), 305–320 (2012)

    CrossRef  MATH  Google Scholar 

  3. Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. of Combinatorial Theory, Series A 114(3), 563–571 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17(1), 1–9 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Alam, M.J., Brandenburg, F.J., Kobourov, S.G.: Straight-line grid drawings of 3-connected 1-planar graphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 83–94. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  6. Angelini, P., Di Battista, G., Didimo, W., Frati, F., Hong, S.H., Kaufmann, M., Liotta, G., Lubiw, A.: Large angle crossing drawings of planar graphs in subquadratic area. In: Márquez, A., Ramos, P., Urrutia, J. (eds.) EGC 2011. LNCS, vol. 7579, pp. 200–209. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  7. Asano, K.: The crossing number of K 1,3,n and K 2,3,n . J. of Graph Theory 10(1), 1–8 (1986)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Auer, C., Bachmaier, C., Brandenburg, F.J., Gleißner, A., Hanauer, K., Neuwirth, D., Reislhuber, J.: Recognizing outer 1-planar graphs in linear time. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 107–118. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  9. Auer, C., Brandenburg, F.J., Gleißner, A., Hanauer, K.: On sparse maximal 2-planar graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 555–556. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  10. 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)

    CrossRef  Google Scholar 

  11. Cheong, O., Har-Peled, S., Kim, H., Kim, H.S.: On the number of edges of fan-crossing free graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 163–173. Springer, Heidelberg (2013)

    Google Scholar 

  12. Dehkordi, H.R., Eades, P.: Every outer-1-plane graph has a right angle crossing drawing. International J. on Computational Geometry and Appl. 22(6), 543–558 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Di Giacomo, E., Didimo, W., Eades, P., Liotta, G.: 2-layer right angle crossing drawings. Algorithmica 68(4), 954–997 (2014)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H.: Area, curve complexity, and crossing resolution of non-planar graph drawings. Theory of Computing Syst. 49(3), 565–575 (2011)

    CrossRef  MATH  Google Scholar 

  15. 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)

    CrossRef  Google Scholar 

  16. 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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Dickerson, M., Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent drawings: Visualizing non-planar diagrams in a planar way. J. of Graph Algorithms and Appl. 9(1), 31–52 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Didimo, W.: Density of straight-line 1-planar graph drawings. Information Processing Letters 113(7), 236–240 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theor. Comput. Sci. 412(39), 5156–5166 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Didimo, W., Liotta, G.: The crossing angle resolution in graph drawing. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory. Springer (2012)

    Google Scholar 

  21. Dujmović, V., Gudmundsson, J., Morin, P., Wolle, T.: Notes on large angle crossing graphs. Chicago J. on Theoretical Computer Science 2011 (2011)

    Google Scholar 

  22. Eades, P., Liotta, G.: Right angle crossing graphs and 1-planarity. Discrete Applied Mathematics 161(7-8), 961–969 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent layered drawings. Algorithmica 47(4), 439–452 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Fox, J., Pach, J., Suk, A.: The number of edges in k-quasi-planar graphs. SIAM J. on Discrete Mathematics 27(1), 550–561 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. Hong, S.H., Eades, P., Katoh, N., Liotta, G., Schweitzer, P., Suzuki, Y.: A linear-time algorithm for testing outer-1-planarity. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 71–82. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  27. 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)

    CrossRef  Google Scholar 

  28. Kaufmann, M., Ueckerdt, T.: The density of fan-planar graphs. CoRR abs/1403.6184 (2014), http://arxiv.org/abs/1403.6184

  29. Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. of Graph Theory 72(1), 30–71 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  30. Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  31. Schaefer, M.: The graph crossing number and its variants: A survey. Electronic J. of Combinatorics 20(2) (2013)

    Google Scholar 

  32. Suzuki, Y.: Re-embeddings of maximum 1-planar graphs. SIAM J. on Discrete Mathematics 24(4), 1527–1540 (2010)

    CrossRef  MATH  Google Scholar 

  33. Valtr, P.: On geometric graphs with no k pairwise parallel edges. Discrete & Computational Geometry 19(3), 461–469 (1998)

    CrossRef  MATH  MathSciNet  Google Scholar 

  34. Garey, M., Johnson, D.: Crossing Number is NP-Complete. SIAM Journal on Algebraic Discrete Methods 4(3), 312–316 (1983), doi:10.1137/0604033

    CrossRef  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Università degli Studi di Perugia, Italy

    Carla Binucci, Emilio Di Giacomo, Walter Didimo & Fabrizio Montecchiani

  2. Università Roma Tre, Italy

    Maurizio Patrignani

  3. Univ. of Crete and Institute of Computer Science-FORTH, Greece

    Ioannis G. Tollis

Authors
  1. Carla Binucci
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  2. Emilio Di Giacomo
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  3. Walter Didimo
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  4. Fabrizio Montecchiani
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  5. Maurizio Patrignani
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  6. Ioannis G. Tollis
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Editor information

Editors and Affiliations

  1. Quinnipiac University, Hamden, CT, USA

    Christian Duncan

  2. National Technical University of Athens, Athens, Greece

    Antonios Symvonis

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Binucci, C., Di Giacomo, E., Didimo, W., Montecchiani, F., Patrignani, M., Tollis, I.G. (2014). Fan-Planar Graphs: Combinatorial Properties and Complexity Results. In: Duncan, C., Symvonis, A. (eds) Graph Drawing. GD 2014. Lecture Notes in Computer Science, vol 8871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45803-7_16

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  • DOI: https://doi.org/10.1007/978-3-662-45803-7_16

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