Skip to main content

Hierarchical Partial Planarity

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10520))

Included in the following conference series:

  • 669 Accesses

Abstract

In this paper we consider graphs whose edges are associated with a degree of importance, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts in which the readability of an edge is proportional to its importance, that is, more important edges have fewer crossings. We formalize this problem and study the case in which there exist three different degrees of importance. We give a polynomial-time testing algorithm when the graph induced by the two most important sets of edges is biconnected. We also discuss interesting relationships with other constrained-planarity problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Angelini, P., Bekos, M.A.: Hierarchical partial planarity. CoRR, 1707.06844, (2017). http://arxiv.org/abs/1707.06844

  2. Angelini, P., Binucci, C., Da Lozzo, G., Didimo, W., Grilli, L., Montecchiani, F., Patrignani, M., Tollis, I.G.: Algorithms and bounds for drawing non-planar graphs with crossing-free subgraphs. Comput. Geom. 50, 34–48 (2015). doi:10.1016/j.comgeo.2015.07.002

    Article  MathSciNet  MATH  Google Scholar 

  3. Angelini, P., et al.: Simultaneous orthogonal planarity. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 532–545. Springer, Cham (2016). doi:10.1007/978-3-319-50106-2_41

    Chapter  Google Scholar 

  4. Angelini, P., Da Lozzo, G., Neuwirth, D.: Advancements on SEFE and partitioned book embedding problems. Theor. Comput. Sci. 575, 71–89 (2015). doi:10.1016/j.tcs.2014.11.016

    Article  MathSciNet  MATH  Google Scholar 

  5. Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32 (2015). doi:10.1145/2629341

    Article  MathSciNet  MATH  Google Scholar 

  6. 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). doi:10.1016/j.jda.2011.12.015.

    Article  MathSciNet  MATH  Google Scholar 

  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). doi:10.7155/jgaa.00388

    Article  MathSciNet  MATH  Google Scholar 

  8. Binucci, C., Brandes, U., Di Battista, G., Didimo, W., Gaertler, M., Palladino, P., Patrignani, M., Symvonis, A., Zweig, K.A.: Drawing trees in a streaming model. Inf. Process. Lett. 112(11), 418–422 (2012). doi:10.1016/j.ipl.2012.02.011

    Article  MathSciNet  MATH  Google Scholar 

  9. Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization, pp. 349–381. Chapman and Hall/CRC, London (2013)

    Google Scholar 

  10. Bläsius, T., Rutter, I.: Disconnectivity and relative positions in simultaneous embeddings. Comput. Geom. 48(6), 459–478 (2015). doi:10.1016/j.comgeo.2015.02.002

    Article  MathSciNet  MATH  Google Scholar 

  11. Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. ACM Trans. Algorithms 12(2), 16:1–16:46 (2016). doi:10.1145/2738054.

    MathSciNet  Google Scholar 

  12. Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. 36(2), 117–130 (2007). doi:10.1016/j.comgeo.2006.05.006

    Article  MathSciNet  MATH  Google Scholar 

  13. Chan, T.M., Frati, F., Gutwenger, C., Lubiw, A., Mutzel, P., Schaefer, M.: Drawing partially embedded and simultaneously planar graphs. J. Graph Algorithms Appl. 19(2), 681–706 (2015). doi:10.7155/jgaa.00375

    Article  MathSciNet  MATH  Google Scholar 

  14. Da Lozzo, G., Rutter, I.: Planarity of streamed graphs. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 153–166. Springer, Cham (2015). doi:10.1007/978-3-319-18173-8_11

    Chapter  Google Scholar 

  15. Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996). doi:10.1007/BF01961541

    Article  MathSciNet  MATH  Google Scholar 

  16. Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996). doi:10.1137/S0097539794280736

    Article  MathSciNet  MATH  Google Scholar 

  17. Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Wismath, S.K.: Planar and quasi-planar simultaneous geometric embedding. Comput. J. 58(11), 3126–3140 (2015). doi:10.1093/comjnl/bxv048

    Article  MATH  Google Scholar 

  18. Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. Graph Algorithms Appl. 9(3), 347–364 (2005). doi:10.7155/jgaa.00113

    Article  MathSciNet  MATH  Google Scholar 

  19. Goodrich, M.T., Pszona, P.: Streamed graph drawing and the file maintenance problem. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 256–267. Springer, Cham (2013). doi:10.1007/978-3-319-03841-4_23

    Chapter  Google Scholar 

  20. Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001). doi:10.1007/3-540-44541-2_8

    Chapter  Google Scholar 

  21. Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. 46(4), 466–492 (2013). doi:10.1016/j.comgeo.2012.07.005

    Article  MathSciNet  MATH  Google Scholar 

  22. Kaufmann, M., Wagner, D. (eds.): Drawing Graphs, Methods and Models. LNCS, vol. 2025. Springer, Heidelberg (2001). doi:10.1016/0095-8956(91)90090-7

    MATH  Google Scholar 

  23. Kratochvíl, J.: String graphs. I. The number of critical nonstring graphs is infinite. J. Comb. Theory, Ser. B 52(1), 53–66 (1991). doi:10.1016/0095-8956(91)90090-7

    Article  MathSciNet  MATH  Google Scholar 

  24. Schaefer, M.: Picking planar edges; or, drawing a graph with a planar subgraph. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 13–24. Springer, Heidelberg (2014). doi:10.1007/978-3-662-45803-7_2

    Google Scholar 

  25. Tamassia, R., Liotta, G.: Graph drawing. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn, pp. 1163–1185. Chapman and Hall/CRC, London (2004). doi:10.1201/9781420035315.ch52

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael A. Bekos .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Angelini, P., Bekos, M.A. (2017). Hierarchical Partial Planarity. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68705-6_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68704-9

  • Online ISBN: 978-3-319-68705-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics