Maximizing Ink in Partial Edge Drawings of k-plane Graphs

  • Matthias Hummel
  • Fabian Klute
  • Soeren Nickel
  • Martin NöllenburgEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)


Partial edge drawing (PED) is a drawing style for non-planar graphs, in which edges are drawn only partially as pairs of opposing stubs on the respective end-vertices. In a PED, by erasing the central parts of edges, all edge crossings and the resulting visual clutter are hidden in the undrawn parts of the edges. In symmetric partial edge drawings (SPEDs), the two stubs of each edge are required to have the same length. It is known that maximizing the ink (or the total stub length) when transforming a straight-line graph drawing with crossings into a SPED is tractable for 2-plane input drawings, but \(\mathsf {NP}\)-hard for unrestricted inputs. We show that the problem remains \(\mathsf {NP}\)-hard even for 3-plane input drawings and establish \(\mathsf {NP}\)-hardness of ink maximization for PEDs of 4-plane graphs. Yet, for k-plane input drawings whose edge intersection graph forms a collection of trees or, more generally, whose intersection graph has bounded treewidth, we present efficient algorithms for computing maximum-ink PEDs and SPEDs. We implemented the treewidth-based algorithms and show a brief experimental evaluation.


  1. 1.
    Abseher, M., Musliu, N., Woltran, S.: htd - a free, open-source framework for (customized) tree decompositions and beyond. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 376–386. Springer, Heidelberg (2017).
  2. 2.
    Becker, R.A., Eick, S.G., Wilks, A.R.: Visualizing network data. IEEE Trans. Vis. Comput. Graph. 1(1), 16–28 (1995). Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008).
  4. 4.
    Binucci, C., Liotta, G., Montecchiani, F., Tappini, A.: Partial edge drawing: homogeneity is more important than crossings and ink. In: Information, Intelligence, Systems Applications (IISA 2016). IEEE (2016).
  5. 5.
    Bruckdorfer, T.: Schematics of Graphs and Hypergraphs. Ph.D. thesis, Universität Tübingen (2015).
  6. 6.
    Bruckdorfer, T., Cornelsen, S., Gutwenger, C., Kaufmann, M., Montecchiani, F., Nöllenburg, M., Wolff, A.: Progress on partial edge drawings. J. Graph Algorithms Appl. 21(4), 757–786 (2017).
  7. 7.
    Bruckdorfer, T., Kaufmann, M.: Mad at edge crossings? Break the edges! In: Kranakis, E., Krizanc, D., Luccio, F. (eds.) FUN 2012. LNCS, vol. 7288, pp. 40–50. Springer, Heidelberg (2012).
  8. 8.
    Bruckdorfer, T., Kaufmann, M., Lauer, A.: A practical approach for 1/4-SHPEDs. In: Information, Intelligence, Systems and Applications (IISA 2015). IEEE (2015).
  9. 9.
    Bruckdorfer, T., Kaufmann, M., Leibßle, S.: PED user study. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 551–553. Springer, Heidelberg (2015).
  10. 10.
    Bruckdorfer, T., Kaufmann, M., Montecchiani, F.: 1-bend orthogonal partial edge drawings. J. Graph Algorithms Appl. 18(1), 111–131 (2014). Scholar
  11. 11.
    Burch, M., Schmauder, H., Panagiotidis, A., Weiskopf, D.: Partial link drawings for nodes, links, and regions of interest. In: Information Visualisation (IV 2014), pp. 53–58 (2014).
  12. 12.
    Burch, M., Vehlow, C., Konevtsova, N., Weiskopf, D.: Evaluating partially drawn links for directed graph edges. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011). LNCS, vol. 7034, pp. 226–237. Springer, Heidelberg (2012).
  13. 13.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms, vol. 3. Springer, Heidelberg (2015).
  14. 14.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (2012)Google Scholar
  15. 15.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Softw. Pract. Exper. 21(11), 1129–1164 (1991).
  16. 16.
    Hummel, M., Klute, F., Nickel, S., Nöllenburg, M.: Maximizing ink inpartial edge drawings of \(k\)-plane graphs. CoRR abs/1908.08905 (2019).
  17. 17.
    Koffka, K.: Principles of Gestalt Psychology. Routledge, Abingdon (1935)Google Scholar
  18. 18.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982). Scholar
  19. 19.
    Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: Di Battista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 248–261. Springer, Heidelberg (1997).
  20. 20.
    Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Comb. Theory Ser. B 36(1), 49–64 (1984).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Algorithms and Complexity GroupTU WienViennaAustria

Personalised recommendations