The Hamiltonian Augmentation Problem and Its Applications to Graph Drawing

  • Emilio Di Giacomo
  • Giuseppe Liotta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)


In this talk we digress about the strict interplay between the graph-theoretic problem of computing a Hamiltonian augmentation of a planar graph G and the graph drawing problem of embedding G onto a given set of points. We review different Hamiltonian augmentation techniques and their impact on different variants of the corresponding graph drawing problem. We also look at universal point sets, simultaneous graph embeddings, and radial graph drawings.


Planar Graph Distinct Point Computational Geometry Hamiltonian Path Curve Complexity 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abellanas, M., Garcia-Lopez, J., Hernández-Peñver, G., Noy, M., Ramos, P.A.: Bipartite embeddings of trees in the plane. Discrete Applied Mathematics 93(2-3), 141–148 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Akiyama, J., Urrutia, J.: Simple alternating path problem. Discrete Mathematics 84, 101–103 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoretical Computer Science 408(2-3), 129–142 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. 36(2), 117–130 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM Journal on Computing 14, 210–223 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chiba, N., Nishizeki, T.: The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. Journal of Algorithms 10, 189–211 (1989)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989)CrossRefGoogle Scholar
  8. 8.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Di Giacomo, E., Didimo, W., Liotta, G.: Radial drawings of graphs: Geometric constraints and trade-offs. Journal of Discrete Algorithms 6(1), 109–124 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. Journal of Graph Algorithms and Applications 12(1), 29–49 (2008)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Computational Geometry 30, 1–23 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. Algorithmica (to appear)Google Scholar
  13. 13.
    Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. IJFCS, Special Issue on Graph Drawing 17(5), 1071–1094 (2006)zbMATHGoogle Scholar
  14. 14.
    Di Giacomo, E., Liotta, G.: Simultaneous embedding of outerplanar graphs, paths, and cycles. International Journal of Computational Geometry and Applications 17(2), 139–160 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Enomoto, H., Miyauchi, M.S.: Embedding graphs into a three page book with O(m logn) crossings of edges over the spine. SIAM J. Discrete Math. 12(3), 337–341 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. Journal of Graph Algorithms and Applications 9(3), 347–364 (2005)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Everett, H., Lazard, S., Liotta, G., Wismath, S.K.: Universal sets of n points for 1-bend drawings of planar graphs with n vertices. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 345–351. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Giordano, F., Liotta, G., Mchedlidze, T., Symvonis, A.: Computing upward topological book embeddings of upward planar digraphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 172–183. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Giordano, F., Liotta, G., Whitesides, S.: Embeddability problems for upward planar digraphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 242–253. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Amer. Math. Monthly 98(2), 165–166 (1991)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Halton, J.H.: On the thickness of graphs of given degree. Information Sciences 54, 219–238 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Applied Mathematics 101, 167–175 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane - a survey. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete & Computational Geometry. Algorithms and Combinatories, vol. 25, pp. 551–570. Springer, Heidelberg (2003)Google Scholar
  24. 24.
    Kaneko, A., Kano, M., Suzuki, K.: Path coverings of two sets of points in the plane. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342. American Mathematical Society (2004)Google Scholar
  25. 25.
    Kaneko, A., Kano, M., Yoshimoto, K.: Alternating hamilton cycles with minimum number of crossing in the plane. International Journal of Computational Geometry & Application 10, 73–78 (2000)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Kaneko, A., Kano, M.: Straight-line embeddings of two rooted trees in the plane. Discrete & Computational Geometry 21(4), 603–613 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kaneko, A., Kano, M., Tokunaga, S.: Straight-line embeddings of three rooted trees in the plane. In: Canadian Conference on Computational Geometry, CCCG 1998 (1998)Google Scholar
  28. 28.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs. Inf. Process. Lett. 92(2), 95–98 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Mchedlidze, T., Symvonis, A.: Crossing-optimal acyclic hamiltonian path completion and its application to upward topological book embeddings. In: Das, S., Uehara, R. (eds.) WALCOM 2009. LNCS, vol. 5431, pp. 250–261. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  31. 31.
    Mchedlidze, T., Symvonis, A.: Crossing-optimal acyclic hp-completion for outerplanar t-digraphs. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 76–85. Springer, Heidelberg (2009)Google Scholar
  32. 32.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph and Combinatorics 17, 717–728 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms (SODA 1990), pp. 138–148 (1990)Google Scholar
  34. 34.
    Sugiyama, K.: Graph Drawing and Applications. World Scientific, Singapore (2002)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Giuseppe Liotta
    • 1
  1. 1.Dip. di Ingegneria Elettronica e dell’InformazioneUniversità degli Studi di Perugia 

Personalised recommendations