Graph Drawings with One Bend and Few Slopes

  • Kolja KnauerEmail author
  • Bartosz Walczak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that \(\lceil \frac{\varDelta }{2}\rceil \) edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree \(\varDelta \geqslant 3\). This matches the obvious lower bound. We also show that \(\lceil \frac{\varDelta }{2}\rceil +1\) edge slopes suffice for drawings of general graphs, improving on the previous bound of \(\varDelta +1\). Furthermore, we improve previous upper bounds on the number of slopes needed for planar drawings of planar and bipartite planar graphs.



We are grateful to Piotr Micek for fruitful discussions.


  1. 1.
    Barát, J., Matoušek, J., Wood, D.R.: Bounded-degree graphs have arbitrarily large geometric thickness. Electron. J. Combin. 13(1), #R3, 14 pp. (2006)Google Scholar
  2. 2.
    Biedl, T., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Czyzowicz, J.: Lattice diagrams with few slopes. J. Combin. Theory, Ser. A 56(1), 96–108 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Czyzowicz, J., Pelc, A., Rival, I., Urrutia, J.: Crooked diagrams with few slopes. Order 7(2), 133–143 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Di Giacomo, E., Liotta, G., Montecchiani, F.: The planar slope number of subcubic graphs. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 132–143. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  6. 6.
    Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. 38(3), 194–212 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dujmović, V., Suderman, M., Wood, D.R.: Graph drawings with few slopes. Comput. Geom. 38(3), 181–193 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dujmović, V., Wood, D.R.: On linear layouts of graphs. Discrete Math. Theor. Comput. Sci. 6(2), 339–358 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Felsner, S., Kaufmann, M., Valtr, P.: Bend-optimal orthogonal graph drawing in the general position model. Comput. Geom. 47(3), 460–468 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    de Fraysseix, H., de Mendez, P.O., Pach, J.: A left-first search algorithm for planar graphs. Discrete Comput. Geom. 13(3–4), 459–468 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Combin. Prob. Comput. 3(2), 233–246 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B., Tesař, M., Vyskočil, T.: The planar slope number of planar partial \(3\)-trees of bounded degree. Graphs Combin. 29(4), 981–1005 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing planar graphs of bounded degree with few slopes. SIAM J. Discrete Math. 27(2), 1171–1183 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Keszegh, B., Pach, J., Pálvölgyi, D., Tóth, G.: Drawing cubic graphs with at most five slopes. Comput. Geom. 40(2), 138–147 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Knauer, K., Micek, P., Walczak, B.: Outerplanar graph drawings with few slopes. Comput. Geom. 47(5), 614–624 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lenhart, W., Liotta, G., Mondal, D., Nishat, R.I.: Planar and plane slope number of partial 2-trees. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 412–423. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  18. 18.
    Liu, Y., Morgana, A., Simeone, B.: A linear algorithm for \(2\)-bend embeddings of planar graphs in the two-dimensional grid. Discrete Appl. Math. 81(1–3), 69–91 (1998)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Misra, J., Gries, D.: A constructive proof of Vizing’s theorem. Inform. Process. Lett. 41(3), 131–133 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mukkamala, P., Pálvölgyi, D.: Drawing cubic graphs with the four basic slopes. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 254–265. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Mukkamala, P., Szegedy, M.: Geometric representation of cubic graphs with four directions. Comput. Geom. 42(9), 842–851 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nomura, K., Tayu, S., Ueno, S.: On the orthogonal drawing of outerplanar graphs. In: Chwa, K.-Y., Munro, J.I. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 300–308. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Pach, J., Pálvölgyi, D.: Bounded-degree graphs can have arbitrarily large slope numbers. Electron. J. Combin. 13(1), #N1, 4 pp. (2006)Google Scholar
  24. 24.
    Vizing, V.G.: Ob otsenke khromaticheskogo klassa \(p\)-grafa (On an estimate of the chromatic class of a \(p\)-graph). Diskret. Analiz 3, 25–30 (1964)MathSciNetGoogle Scholar
  25. 25.
    Wade, G.A., Chu, J.H.: Drawability of complete graphs using a minimal slope set. Comput. J. 37(2), 139–142 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Aix-Marseille Université, CNRS, LIF UMR 7279MarseilleFrance
  2. 2.Theoretical Computer Science Department, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland

Personalised recommendations