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A Universal Slope Set for 1-Bend Planar Drawings

Authors Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, Fabrizio Montecchiani

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Patrizio Angelini
Michael A. Bekos
Giuseppe Liotta
Fabrizio Montecchiani

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Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, and Fabrizio Montecchiani. A Universal Slope Set for 1-Bend Planar Drawings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).
  • Slope number
  • 1-bend drawings
  • planar graphs
  • angular resolution


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  1. Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, and Fabrizio Montecchiani. Universal slope sets for 1-bend planar drawings. CoRR, 1703.04283, 2017. Google Scholar
  2. János Barát, Jirí Matoušek, and David R. Wood. Bounded-degree graphs have arbitrarily large geometric thickness. Electr. J. Comb., 13(1), 2006. Google Scholar
  3. Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Robert Krug. Planar octilinear drawings with one bend per edge. J. Graph Algorithms Appl., 19(2):657-680, 2015. URL:
  4. Michael A. Bekos, Michael Kaufmann, and Robert Krug. On the total number of bends for planar octilinear drawings. In LATIN, volume 9644 of LNCS, pages 152-163. Springer, 2016. URL:
  5. Therese C. Biedl and Goos Kant. A better heuristic for orthogonal graph drawings. Comput. Geom., 9(3):159-180, 1998. URL:
  6. Thomas Bläsius, Marcus Krug, Ignaz Rutter, and Dorothea Wagner. Orthogonal graph drawing with flexibility constraints. Algorithmica, 68(4):859-885, 2014. URL:
  7. Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inflexible edges. Comput. Geom., 55:26-40, 2016. Google Scholar
  8. Hans L. Bodlaender and Gerard Tel. A note on rectilinearity and angular resolution. J. Graph Algorithms Appl., 8:89-94, 2004. Google Scholar
  9. Nicolas Bonichon, Bertrand Le Saëc, and Mohamed Mosbah. Optimal area algorithm for planar polyline drawings. In WG, volume 2573 of LNCS, pages 35-46. Springer, 2002. Google Scholar
  10. Hubert de Fraysseix, Patrice Ossona de Mendez, and Pierre Rosenstiehl. On triangle contact graphs. Combinatorics, Probability & Computing, 3:233-246, 1994. URL:
  11. Hubert de Fraysseix, János Pach, and Richard Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. URL:
  12. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999. Google Scholar
  13. Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. The planar slope number of subcubic graphs. In LATIN, volume 8392 of LNCS, pages 132-143. Springer, 2014. URL:
  14. Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. Drawing outer 1-planar graphs with few slopes. J. Graph Algorithms Appl., 19(2):707-741, 2015. URL:
  15. Vida Dujmović, David Eppstein, Matthew Suderman, and David R. Wood. Drawings of planar graphs with few slopes and segments. Comput. Geom., 38(3):194-212, 2007. Google Scholar
  16. Christian A. Duncan, David Eppstein, Michael T. Goodrich, Stephen G. Kobourov, and Martin Nöllenburg. Drawing trees with perfect angular resolution and polynomial area. Discrete & Computational Geometry, 49(2):157-182, 2013. Google Scholar
  17. Christian A. Duncan and Stephen G. Kobourov. Polar coordinate drawing of planar graphs with good angular resolution. J. Graph Algorithms Appl., 7(4):311-333, 2003. Google Scholar
  18. Stephane Durocher and Debajyoti Mondal. Trade-offs in planar polyline drawings. In GD, volume 8871 of LNCS, pages 306-318. Springer, 2014. Google Scholar
  19. Michael Formann, Torben Hagerup, James Haralambides, Michael Kaufmann, Frank Thomson Leighton, Antonios Symvonis, Emo Welzl, and Gerhard J. Woeginger. Drawing graphs in the plane with high resolution. SIAM J. Comput., 22(5):1035-1052, 1993. URL:
  20. Ashim Garg and Roberto Tamassia. Planar drawings and angular resolution: Algorithms and bounds (extended abstract). In ESA, volume 855 of LNCS, pages 12-23. Springer, 1994. Google Scholar
  21. Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput., 31(2):601-625, 2001. URL:
  22. Carsten Gutwenger and Petra Mutzel. Planar polyline drawings with good angular resolution. In GD, volume 1547 of LNCS, pages 167-182. Springer, 1998. Google Scholar
  23. Carsten Gutwenger and Petra Mutzel. A linear time implementation of SPQR-trees. In GD, volume 1984 of LNCS, pages 77-90. Springer, 2000. URL:
  24. Udo Hoffmann. On the complexity of the planar slope number problem. J. Graph Algorithms Appl., 21(2):183-193, 2017. Google Scholar
  25. Vít Jelínek, Eva Jelínková, Jan Kratochvíl, Bernard Lidický, Marek Tesar, and Tomás Vyskocil. The planar slope number of planar partial 3-trees of bounded degree. Graphs and Comb., 29(4):981-1005, 2013. URL:
  26. Michael Jünger and Petra Mutzel, editors. Graph Drawing Software. Springer, 2004. Google Scholar
  27. Goos Kant. Drawing planar graphs using the lmc-ordering (extended abstract). In FOCS, pages 101-110. IEEE Computer Society, 1992. URL:
  28. Goos Kant. Hexagonal grid drawings. In WG, volume 657 of LNCS, pages 263-276. Springer, 1992. URL:
  29. Goos Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16(1):4-32, 1996. URL:
  30. Balázs Keszegh, János Pach, and Dömötör Pálvölgyi. Drawing planar graphs of bounded degree with few slopes. SIAM J. Discrete Math., 27(2):1171-1183, 2013. URL:
  31. Kolja Knauer and Bartosz Walczak. Graph drawings with one bend and few slopes. In LATIN, volume 9644 of LNCS, pages 549-561. Springer, 2016. URL:
  32. Kolja B. Knauer, Piotr Micek, and Bartosz Walczak. Outerplanar graph drawings with few slopes. Comput. Geom., 47(5):614-624, 2014. URL:
  33. William Lenhart, Giuseppe Liotta, Debajyoti Mondal, and Rahnuma Islam Nishat. Planar and plane slope number of partial 2-trees. In GD, volume 8242 of LNCS, pages 412-423. Springer, 2013. URL:
  34. Yanpei Liu, Aurora Morgana, and Bruno Simeone. A linear algorithm for 2-bend embeddings of planar graphs in the two-dimensional grid. Discrete Applied Mathematics, 81(1-3):69-91, 1998. URL:
  35. Padmini Mukkamala and Dömötör Pálvölgyi. Drawing cubic graphs with the four basic slopes. In GD, volume 7034 of LNCS, pages 254-265. Springer, 2011. URL:
  36. Martin Nöllenburg. Automated drawings of metro maps. Technical Report 2005-25, Fakultät für Informatik, Universität Karlsruhe, 2005. Google Scholar
  37. Martin Nöllenburg and Alexander Wolff. Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE Trans. Vis. Comput. Graph., 17(5):626-641, 2011. URL:
  38. Jonathan M. Stott, Peter Rodgers, Juan Carlos Martinez-Ovando, and Stephen G. Walker. Automatic metro map layout using multicriteria optimization. IEEE Trans. Vis. Comput. Graph., 17(1):101-114, 2011. URL:
  39. Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421-444, 1987. URL:
  40. Roberto Tamassia, editor. Handbook on Graph Drawing and Visualization. CRC Press, 2013. Google Scholar
  41. Greg A. Wade and Jiang-Hsing Chu. Drawability of complete graphs using a minimal slope set. Comput. J., 37(2):139-142, 1994. URL:
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