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Planar Drawings of Fixed-Mobile Bigraphs

  • Michael A. Bekos
  • Felice De Luca
  • Walter Didimo
  • Tamara Mchedlidze
  • Martin Nöllenburg
  • Antonios Symvonis
  • Ioannis G. Tollis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)

Abstract

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given \(k \ge 0\), a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For \(k=0\) and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of “layered” 1-bend drawings.

References

  1. 1.
    Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoret. Comput. Sci. 408(2–3), 129–142 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barth, L., Gemsa, A., Niedermann, B., Nöllenburg, M.: On the readability of boundary labeling. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 515–527. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-27261-0_42 CrossRefGoogle Scholar
  3. 3.
    Bekos, M.A., Cornelsen, S., Fink, M., Hong, S., Kaufmann, M., Nöllenburg, M., Rutter, I., Symvonis, A.: Many-to-one boundary labeling with backbones. J. Graph Algorithms Appl. 19(3), 779–816 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bekos, M.A., De Luca, F., Didimo, W., Mchedlidze, T., Nöllenburg, M., Symvonis, A., Tollis., I.: Planar drawings of fixed-mobile bigraphs. CoRR 1708.09238 (2017)Google Scholar
  5. 5.
    Bekos, M.A., Kaufmann, M., Symvonis, A., Wolff, A.: Boundary labeling: models and efficient algorithms for rectangular maps. Comput. Geom. 36(3), 215–236 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Biedl, T.C.: Drawing planar partitions I: LL-drawings and LH-drawings. In: Janardan, R. (ed.) Computational Geometry (SoCG 1998), pp. 287–296. ACM (1998)Google Scholar
  7. 7.
    Biedl, T., Kaufmann, M., Mutzel, P.: Drawing planar partitions II: HH-drawings. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 124–136. Springer, Heidelberg (1998).  https://doi.org/10.1007/10692760_11 CrossRefGoogle Scholar
  8. 8.
    Brandes, U., Erten, C., Estrella-Balderrama, A., Fowler, J.J., Frati, F., Geyer, M., Gutwenger, C., Hong, S., Kaufmann, M., Kobourov, S.G., Liotta, G., Mutzel, P., Symvonis, A.: Colored simultaneous geometric embeddings and universal pointsets. Algorithmica 60(3), 569–592 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chambers, E.W., Eppstein, D., Goodrich, M.T., Löffler, M.: Drawing graphs in the plane with a prescribed outer face and polynomial area. J. Graph Algorithms Appl. 16(2), 243–259 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1999)zbMATHGoogle Scholar
  11. 11.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. J. Graph Algorithms Appl. 12(1), 29–49 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. Algorithmica 57(4), 796–818 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Planar drawings of higher-genus graphs. J. Graph Algorithms Appl. 15(1), 7–32 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fößmeier, U., Kaufmann, M.: Nice drawings for planar bipartite graphs. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds.) CIAC 1997. LNCS, vol. 1203, pp. 122–134. Springer, Heidelberg (1997).  https://doi.org/10.1007/3-540-62592-5_66 CrossRefGoogle Scholar
  15. 15.
    Goaoc, X., Kratochvíl, J., Okamoto, Y., Shin, C., Spillner, A., Wolff, A.: Untangling a planar graph. Discrete Comput. Geom. 42(4), 542–569 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Halperin, D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chap. 24, pp. 529–562. CRC Press LLC, Boca Raton (2004)Google Scholar
  17. 17.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1972)Google Scholar
  18. 18.
    Hong, S.H., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discrete Appl. Math. 156(12), 2368–2380 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ito, T., Misue, K., Tanaka, J.: Sphere anchored map: a visualization technique for bipartite graphs in 3D. In: Jacko, J.A. (ed.) HCI 2009. LNCS, vol. 5611, pp. 811–820. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02577-8_89 CrossRefGoogle Scholar
  20. 20.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44969-8 zbMATHGoogle Scholar
  21. 21.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kindermann, P., Niedermann, B., Rutter, I., Schaefer, M., Schulz, A., Wolff, A.: Multi-sided boundary labeling. Algorithmica 76(1), 225–258 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lin, C.: Crossing-free many-to-one boundary labeling with hyperleaders. In: IEEE Pacific Visualization Symposium PacificVis 2010, Taipei, Taiwan, 2–5 March 2010, pp. 185–192. IEEE Computer Society (2010)Google Scholar
  24. 24.
    Mchedlidze, T., Nöllenburg, M., Rutter, I.: Extending convex partial drawings of graphs. Algorithmica 76(1), 47–67 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  26. 26.
    Misue, K.: Anchored map: Graph drawing technique to support network mining. IEICE Trans. 91–D(11), 2599–2606 (2008)CrossRefGoogle Scholar
  27. 27.
    Misue, K., Zhou, Q.: Drawing semi-bipartite graphs in anchor+matrix style. In: Information Visualisation (IV 2011), London, UK, 13–15 July 2011, pp. 26–31. IEEE Computer Society (2011)Google Scholar
  28. 28.
    Neyer, G.: Map labeling with application to graph drawing. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs. LNCS, vol. 2025, pp. 247–273. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44969-8_10 CrossRefGoogle Scholar
  29. 29.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17(4), 717–728 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Patrignani, M.: On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1070 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Patrignani, M.: On extending a partial straight-line drawing. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 380–385. Springer, Heidelberg (2006).  https://doi.org/10.1007/11618058_34 CrossRefGoogle Scholar
  32. 32.
    Purchase, H.C.: Metrics for graph drawing aesthetics. J. Vis. Lang. Comput. 13(5), 501–516 (2002)CrossRefGoogle Scholar
  33. 33.
    Purchase, H.C., Carrington, D.A., Allder, J.: Empirical evaluation of aesthetics-based graph layout. Empirical Softw. Eng. 7(3), 233–255 (2002)CrossRefzbMATHGoogle Scholar
  34. 34.
    Tamassia, R. (ed.): Handbook on Graph Drawing and Visualization. Chapman and Hall/CRC, Boca Raton (2013)zbMATHGoogle Scholar
  35. 35.
    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, Boca Raton (2004)Google Scholar
  36. 36.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13(3), 743–768 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wolff, A., Strijk, T.: The map-labeling bibliography (1996). http://i11www.ira.uka.de/map-labeling/bibliography

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Michael A. Bekos
    • 1
  • Felice De Luca
    • 2
  • Walter Didimo
    • 2
  • Tamara Mchedlidze
    • 3
  • Martin Nöllenburg
    • 4
  • Antonios Symvonis
    • 5
  • Ioannis G. Tollis
    • 6
  1. 1.University of TübingenTübingenGermany
  2. 2.Università degli Studi di PerugiaPerugiaItaly
  3. 3.Karlsruhe Institute of TechnologyKarlsruheGermany
  4. 4.TU WienViennaAustria
  5. 5.National Technical University of AthensAthensGreece
  6. 6.University of CreteHeraklionGreece

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