Drawing Graphs on Few Lines and Few Planes

  • Steven Chaplick
  • Krzysztof Fleszar
  • Fabian Lipp
  • Alexander RavskyEmail author
  • Oleg Verbitsky
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows.

  • We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values.

  • We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing).

  • We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.


  1. 1.
    Akiyama, J., Era, H., Gervacio, S.V., Watanabe, M.: Path chromatic numbers of graphs. J. Graph Theory 13(5), 571–573 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akiyama, J., Exoo, G., Harary, F.: Covering and packing ingraphs III: cyclic and acyclic invariants. Math. Slovaca 30, 405–417 (1980)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bannister, M.J., Devanny, W.E., Dujmović, V., Eppstein, D., Wood, D.R.: Track layouts, layered path decompositions, and leveled planarity (2015).
  4. 4.
    Battista, G.D., Frati, F., Pach, J.: On the queue number of planar graphs. SIAM J. Comput. 42(6), 2243–2285 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bollobás, B.: Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability, 1st edn. Cambridge University Press, Cambridge (1986)zbMATHGoogle Scholar
  6. 6.
    Broere, I., Mynhardt, C.M.: Generalized colorings of outerplanar and planar graphs. In: Proceedings of the 5th International Conference Graph Theory Applications Algorithms and Computer Science, Kalamazoo, MI, pp. 151–161 (1984, 1985)Google Scholar
  7. 7.
    Brooks, R.L.: On colouring the nodes of a network. Math. Proc. Camb. Philos. Soc. 37, 194–197 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chan, T.M., Goodrich, M.T., Kosaraju, S.R., Tamassia, R.: Optimizing area and aspect ratio in straight-line orthogonal tree drawings. Comput. Geom. Theory Appl. 23, 153–162 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: The complexity of drawing graphs on few lines and few planes (2016).
  10. 10.
    Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: Drawing graphs on few lines and few planes (2016).
  11. 11.
    Chartrand, G., Kronk, H.V.: The point-arboricity of planar graphs. J. Lond. Math. Soc. 44, 612–616 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, Z., He, X.: Parallel complexity of partitioning a planar graph into vertex-induced forests. Discrete Appl. Math. 69(1–2), 183–198 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dujmović, V.: Graph layouts via layered separators. J. Comb. Theory Ser. B 110, 79–89 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. Theory Appl. 38(3), 194–212 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM J. Comput. 34(3), 553–579 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dujmović, V., Whitesides, S.: Three-dimensional drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, chap. 14, pp. 455–488. CRC Press, Boca Raton (2013)Google Scholar
  17. 17.
    Durocher, S., Mondal, D.: Drawing plane triangulations with few segments. In: Proceedings of the Canadian Conference Computational Geometry (CCCG 2014), pp. 40–45 (2014).
  18. 18.
    Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl. 7(4), 363–398 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Flum, J., Grohe, M.: Parametrized Complexity Theory. Springer, Berlin (2006)zbMATHGoogle Scholar
  20. 20.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Goddard, W.: Acyclic colorings of planar graphs. Discrete Math. 91(1), 91–94 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grünbaum, B., Walther, H.: Shortness exponents of families of graphs. J. Comb. Theory Ser. A 14(3), 364–385 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hanani, H.: The existence and construction of balanced incomplete block designs. Ann. Math. Stat. 32, 361–386 (1961). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Harary, F.: Covering and packing in graphs I. Ann. N.Y. Acad. Sci. 175, 198–205 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kostochka, A., Melnikov, L.: On a lower bound for the isoperimetric number of cubic graphs. In: Proceedings of the 3rd International Petrozavodsk Conference Probabilistic Methods in Discrete Mathematics, pp. 251–265. TVP, Moskva, VSP, Utrecht (1993)Google Scholar
  26. 26.
    Krein, M., Milman, D.: On extreme points of regular convex sets. Studia Math. 9, 133–138 (1940)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Matsumoto, M.: Bounds for the vertex linear arboricity. J. Graph Theory 14(1), 117–126 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pach, J., Thiele, T., Tóth, G.: Three-dimensional grid drawings of graphs. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 47–51. Springer, Heidelberg (1997). doi: 10.1007/3-540-63938-1_49 CrossRefGoogle Scholar
  29. 29.
    Poh, K.S.: On the linear vertex-arboricity of a planar graph. J. Graph Theory 14(1), 73–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Raspaud, A., Wang, W.: On the vertex-arboricity of planar graphs. Eur. J. Comb. 29(4), 1064–1075 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ravsky, A., Verbitsky, O.: On collinear sets in straight-line drawings. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 295–306. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-25870-1_27. CrossRefGoogle Scholar
  32. 32.
    Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst., 1–22 (2015, has appeared online)Google Scholar
  33. 33.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the 1st ACM-SIAM Symposium Discrete Algorithms (SODA 1990), pp. 138–148 (1990)Google Scholar
  34. 34.
    Schulz, A.: Drawing graphs with few arcs. J. Graph Algorithms Appl. 19(1), 393–412 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wang, J.: On point-linear arboricity of planar graphs. Discrete Math. 72(1–3), 381–384 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wikipedia: Krein-Milman theorem. Accessed 21 Apr 2016
  37. 37.
    Wood, D.R.: Three-dimensional graph drawing. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, pp. 1–7. Springer, Boston (2008)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Krzysztof Fleszar
    • 1
  • Fabian Lipp
    • 1
  • Alexander Ravsky
    • 2
    Email author
  • Oleg Verbitsky
    • 3
  • Alexander Wolff
    • 1
  1. 1.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  2. 2.Pidstryhach Institute for Applied Problems of Mechanics and MathematicsNational Academy of Sciences of UkraineLvivUkraine
  3. 3.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations