Advertisement

Algorithmica

, Volume 81, Issue 6, pp 2527–2556 | Cite as

Universal Slope Sets for 1-Bend Planar Drawings

  • Patrizio Angelini
  • Michael A. Bekos
  • Giuseppe Liotta
  • Fabrizio MontecchianiEmail author
Article
  • 50 Downloads

Abstract

We prove that every set of \(\varDelta -1\) slopes is 1-bend universal for the planar graphs with maximum vertex degree \(\varDelta \). This means that any planar graph with maximum degree \(\varDelta \) admits a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges can be chosen in any given set of \(\varDelta -1\) slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of \(\frac{3}{2} (\varDelta -1)\) on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of \(O(\varDelta )\) slopes that may vary depending on the input graph; thirdly, while these algorithms typically minimize the slopes at the expenses of constructing drawings with poor angular resolution, we can compute drawings whose angular resolution is at least \(\frac{\pi }{\varDelta -1}\), which is worst-case optimal up to a factor of \(\frac{3}{4}\). Our proofs are constructive and give rise to a linear-time drawing algorithm.

Keywords

Graph drawing Slope number 1-Bend planar drawings 

Notes

Acknowledgements

This work started at the 19th Korean Workshop on Computational Geometry. We wish to thank the organizers and the participants for creating a pleasant and stimulating atmosphere and in particular Fabian Lipp and Boris Klemz for useful discussions. We also gratefully thank the anonymous reviewers for their useful comments.

References

  1. 1.
    Barát, J., Matoušek, J., Wood, D.R.: Bounded-degree graphs have arbitrarily large geometric thickness. Electron. J. Comb. 13(1), 3 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bekos, M.A., Gronemann, M., Kaufmann, M., Krug, R.: Planar octilinear drawings with one bend per edge. J. Graph Algorithms Appl. 19(2), 657–680 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bekos, M.A., Kaufmann, M., Krug, R.: On the total number of bends for planar octilinear drawings. J. Graph Algorithms Appl. 21(4), 709–730 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bertolazzi, P., Di Battista, G., Mannino, C., Tamassia, R.: Optimal upward planarity testing of single-source digraphs. SIAM J. Comput. 27(1), 132–169 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bläsius, T., Krug, M., Rutter, I., Wagner, D.: Orthogonal graph drawing with flexibility constraints. Algorithmica 68(4), 859–885 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bläsius, T., Lehmann, S., Rutter, I.: Orthogonal graph drawing with inflexible edges. Comput. Geom. 55, 26–40 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bodlaender, H.L., Tel, G.: A note on rectilinearity and angular resolution. J. Graph Algorithms Appl. 8, 89–94 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bonichon, N., Saëc, B. L., Mosbah, M.: Optimal area algorithm for planar polyline drawings. In: Kucera, L. (ed.) WG, vol. 2573 of LNCS, pp. 35–46. Springer, Berlin (2002)Google Scholar
  10. 10.
    Czyzowicz, J.: Lattice diagrams with few slopes. J. Comb. Theory Ser. A 56(1), 96–108 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Czyzowicz, J., Pelc, A., Rival, I.: Drawing orders with few slopes. Discrete Math. 82(3), 233–250 (1990)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Czyzowicz, J., Pelc, A., Rival, I., Urrutia, J.: Crooked diagrams with few slopes. Order 7(2), 133–143 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Comb. Probab. Comput. 3, 233–246 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    Di Giacomo, E., Liotta, G., Montecchiani, F.: Drawing outer 1-planar graphs with few slopes. J. Graph Algorithms Appl. 19(2), 707–741 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Di Giacomo, E., Liotta, G., Montecchiani, F.: 1-bend upward planar drawings of SP-digraphs. In: Hu, Y., Nöllenburg, M. (eds.), Graph Drawing, vol. 9801 of LNCS, pp. 123–130. Springer, Berlin (2016)Google Scholar
  18. 18.
    Di Giacomo, E., Liotta, G., Montecchiani, F.: Drawing subcubic planar graphs with four slopes and optimal angular resolution. Theor. Comput. Sci. 714, 51–73 (2018)MathSciNetzbMATHGoogle Scholar
  19. 19.
    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)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Dujmović, V., Suderman, M., Wood, D.R.: Graph drawings with few slopes. Comput. Geom. 38(3), 181–193 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Drawing trees with perfect angular resolution and polynomial area. Discrete Comput. Geom. 49(2), 157–182 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Duncan, C.A., Kobourov, S.G.: Polar coordinate drawing of planar graphs with good angular resolution. J. Graph Algorithms Appl. 7(4), 311–333 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Durocher, S., Mondal, D.: On balanced +-contact representations. In: Graph Drawing, vol. 8242 of LNCS, pp. 143–154. Springer, Berlin (2013)Google Scholar
  24. 24.
    Durocher, S., Mondal, D.: Trade-offs in planar polyline drawings. In: Duncan, C.A., Symvonis, A. (eds.) Graph Drawing, vol. 8871 of LNCS, pp. 306–318. Springer, Berlin (2014)Google Scholar
  25. 25.
    Felsner, S., Kaufmann, M., Valtr, P.: Bend-optimal orthogonal graph drawing in the general position model. Comput. Geom. 47(3), 460–468 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Formann, M., Hagerup, T., Haralambides, J., Kaufmann, M., Leighton, F.T., Symvonis, A., Welzl, E., Woeginger, G.J.: Drawing graphs in the plane with high resolution. SIAM J. Comput. 22(5), 1035–1052 (1993)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Garg, A., Tamassia, R.: Planar drawings and angular resolution: algorithms and bounds (extended abstract). In: van Leeuwen, J. (ed.) ESA, vol. 855 of LNCS, pp. 12–23. Springer, Berlin (1994)Google Scholar
  28. 28.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Gronemann, M.: Bitonic \(st\)-orderings of biconnected planar graphs. In: Graph Drawing, vol. 8871 of LNCS, pp. 162–173. Springer, Berlin (2014)Google Scholar
  30. 30.
    Gutwenger, C., Mutzel, P.: Planar polyline drawings with good angular resolution. In: Graph Drawing, vol. 1547 of LNCS, pp. 167–182. Springer, Berlin (1998)Google Scholar
  31. 31.
    Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed), Graph Drawing, vol. 1984 of LNCS, pp. 77–90. Springer, Berlin (2000)Google Scholar
  32. 32.
    Hoffmann, U.: On the complexity of the planar slope number problem. J. Graph Algorithms Appl. 21(2), 183–193 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B., Tesar, M., Vyskocil, T.: The planar slope number of planar partial 3-trees of bounded degree. Graphs Comb. 29(4), 981–1005 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Jünger, M., Mutzel, P. (eds.): Graph Drawing Software. Springer, Berlin (2004)Google Scholar
  35. 35.
    Kant, G.: Drawing planar graphs using the lmc-ordering (extended abstract). In: FOCS, pp. 101–110. IEEE Computer Society (1992)Google Scholar
  36. 36.
    Kant, G.: Hexagonal grid drawings. In: Mayr, E.W. (ed.) WG, vol. 657 of LNCS, pp. 263–276. Springer, Berlin (1992)Google Scholar
  37. 37.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kant, G., Bodlaender, H.L.: Triangulating planar graphs while minimizing the maximum degree. Inf. Comput. 135(1), 1–14 (1997)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing planar graphs of bounded degree with few slopes. SIAM J. Discrete Math. 27(2), 1171–1183 (2013)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Knauer, K., Walczak, B.: Graph drawings with one bend and few slopes. In: Kranakis, E., Navarro, G., Chávez, E. (eds.), LATIN, vol. 9644 of LNCS, pp. 549–561. Springer, Berlin (2016)Google Scholar
  41. 41.
    Knauer, K.B., Micek, P., Walczak, B.: Outerplanar graph drawings with few slopes. Comput. Geom. 47(5), 614–624 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Lenhart, W., Liotta, G., Mondal, D., Nishat, R.I.: Planar and plane slope number of partial 2-trees. In: Wismath, S.K., Wolff, A. (eds.) Graph Drawing, vol. 8242 of LNCS, pp. 412–423. Springer, Berlin (2013)Google Scholar
  43. 43.
    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
  44. 44.
    Mukkamala, P., Pálvölgyi, D.: Drawing cubic graphs with the four basic slopes. In: van Kreveld, M.J., Speckmann, B. (eds.) Graph Drawing, vol. 7034 of LNCS, pp. 254–265. Springer, Berlin (2011)Google Scholar
  45. 45.
    Nöllenburg, M.: Automated drawings of metro maps. Technical Report 2005-25, Fakultät für Informatik, Universität Karlsruhe (2005)Google Scholar
  46. 46.
    Nöllenburg, M., Wolff, A.: Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE Trans. Vis. Comput. Graph. 17(5), 626–641 (2011)Google Scholar
  47. 47.
    Stott, J.M., Rodgers, P., Martinez-Ovando, J.C., Walker, S.G.: Automatic metro map layout using multicriteria optimization. IEEE Trans. Vis. Comput. Graph. 17(1), 101–114 (2011)Google Scholar
  48. 48.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization. CRC Press, Amsterdam (2013)Google Scholar
  50. 50.
    Wade, G.A., Chu, J.: Drawability of complete graphs using a minimal slope set. Comput. J. 37(2), 139–142 (1994)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Wilhelm-Schickhard-Institut für InformatikUniversität Tübingen TübingenGermany
  2. 2.Dipartimento di IngegneriaUniversity of Perugia PerugiaItaly

Personalised recommendations