Convexity-Increasing Morphs of Planar Graphs

  • Linda Kleist
  • Boris KlemzEmail author
  • Anna Lubiw
  • Lena Schlipf
  • Frank Staals
  • Darren Strash
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)


We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of a 3-connected graph G, we show how to morph the drawing to one with convex faces while maintaining planarity at all times. Furthermore, the morph is convexity increasing, meaning that angles of inner faces never change from convex to reflex. We give a polynomial time algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines.


Convex Drawings Planar Straight-line Drawing Horizontal Morph Vertical Morph Morphic Planes 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank André Schulz for helpful discussions on generalizations of Tutte’s algorithm. This work was begun at Dagstuhl workshop 17072, “Applications of Topology to the Analysis of 1-Dimensional Objects.” We thank Dagstuhl, the organizers, and the other participants for a stimulating workshop. In particular, we thank Carola Wenk and Regina Rotmann for joining some of our discussions, and Irina Kostitsyna for contributing many valuable ideas.


  1. 1.
    Aichholzer, O., Aloupis, G., Demaine, E.D., Demaine, M.L., Dujmovic, V., Hurtado, F., Lubiw, A., Rote, G., Schulz, A., Souvaine, D.L., Winslow, A.: Convexifying polygons without losing visibilities. In: Canadian Conference on Computational Geometry (CCCG) (2011)Google Scholar
  2. 2.
    Alamdari, S., Angelini, P., Barrera-Cruz, F., Chan, T.M., Da Lozzo, G., Di Battista, G., Frati, F., Haxell, P., Lubiw, A., Patrignani, M., Roselli, V., Singla, S., Wilkinson, B.T.: How to morph planar graph drawings. SIAM J. Comput. 46(2), 29 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Angelini, P., Da Lozzo, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V.: Optimal morphs of convex drawings. In: Arge, L., Pach, J. (eds.) Proceedings of the 31st International Symposium on Computational Geometry (SoCG 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 34, pp. 126–140. Dagstuhl, Wadern (2015)Google Scholar
  4. 4.
    Appelle, S.: Perception and discrimination as a function of stimulus orientation: the “oblique effect” in man and animals. Psychol. Bull. 78(4), 266 (1972)CrossRefGoogle Scholar
  5. 5.
    Cairns, S.: Deformations of plane rectilinear complexes. Am. Math. Monthly 51(5), 247–252 (1944)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cantarella, J.H., Demaine, E.D., Iben, H.N., O’Brien, J.F.: An energy-driven approach to linkage unfolding. In: Proceedings of the 20th Annual Symposium on Computational Geometry (SoCG), pp. 134–143. ACM (2004)Google Scholar
  7. 7.
    Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discret. Comput. Geom. 30, 205–239 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61(2–3), 175–198 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eiglsperger, M., Fekete, S.P., Klau, G.W.: Orthogonal graph drawing. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs. LNCS, pp. 121–171. Springer, Heidelberg (2001). Scholar
  10. 10.
    Floater, M.S.: Parametric Tilings and scattered data approximation. Int. J. Shape Model. 4(03n04), 165–182 (1998)CrossRefGoogle Scholar
  11. 11.
    Floater, M.S., Gotsman, C.: How to morph tilings injectively. J. Comput. Appl. Math. 101(1–2), 117–129 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gortler, S.J., Gotsman, C., Thurston, D.: Discrete one-forms on meshes and applications to 3D mesh parameterization. Comput. Aided Geomet. Des. 23(2), 83–112 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gotsman, C., Surazhsky, V.: Guaranteed intersection-free polygon morphing. Comput. Graph. 25(1), 67–75 (2001)CrossRefGoogle Scholar
  14. 14.
    Hong, S.-H., Nagamochi, H.: Convex drawings of hierarchical planar graphs and clustered planar graphs. J. Discrete Algorithms 8(3), 282–295 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Iben, H.N., O’Brien, J.F., Demaine, E.D.: Refolding planar polygons. Discret. Comput. Geomet. 41(3), 444–460 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kleist, L., Klemz, B., Lubiw, A., Schlipf, L., Staals, F., Strash, D.: Convexity-increasing morphs of planar graphs. CoRR, abs/1802.06579 (2018)Google Scholar
  17. 17.
    Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pp. 296–303. ACM (2014)Google Scholar
  18. 18.
    Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM J. Numer. Anal. 16(2), 346–358 (1979)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Purchase, H.C., Hoggan, E., Görg, C.: How important is the “Mental map”? – An empirical investigation of a dynamic graph layout algorithm. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 184–195. Springer, Heidelberg (2007). Scholar
  21. 21.
    Purchase, H.C., Pilcher, C., Plimmer, B.: Graph drawing aesthetics–created by users, not algorithms. IEEE Trans. Vis. Comput. Graph. 18(1), 81–92 (2012)CrossRefGoogle Scholar
  22. 22.
    Rahman, S.: Convex graph drawing. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, pp. 1–7. Springer, Heidelberg (2015). Scholar
  23. 23.
    Ribó Mor, A., Rote, G., Schulz, A.: Small grid embeddings of 3-polytopes. Discret. Comput. Geomet. 45(1), 65–87 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Thomassen, C.: Deformations of plane graphs. J. Combin. Theory, Ser. B 34(3), 244–257 (1983)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tutte, W.T.: How to draw a graph. Proc. Lond. Math. Soc. 3(1), 743–767 (1963)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Linda Kleist
    • 1
  • Boris Klemz
    • 2
    Email author
  • Anna Lubiw
    • 3
  • Lena Schlipf
    • 4
  • Frank Staals
    • 5
  • Darren Strash
    • 6
  1. 1.Technische Universität BerlinBerlinGermany
  2. 2.Freie Universität BerlinBerlinGermany
  3. 3.University of WaterlooWaterlooCanada
  4. 4.FernUniversität in HagenHagenGermany
  5. 5.Utrecht UniversityUtrechtThe Netherlands
  6. 6.Hamilton CollegeClintonUSA

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