Skip to main content

The Hamiltonian Augmentation Problem and Its Applications to Graph Drawing

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5942))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Akiyama, J., Urrutia, J.: Simple alternating path problem. Discrete Mathematics 84, 101–103 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoretical Computer Science 408(2-3), 129–142 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM Journal on Computing 14, 210–223 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989)

    Article  Google Scholar 

  8. de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  11. Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Computational Geometry 30, 1–23 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Halton, J.H.: On the thickness of graphs of given degree. Information Sciences 54, 219–238 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Applied Mathematics 101, 167–175 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    MATH  MathSciNet  Google Scholar 

  26. Kaneko, A., Kano, M.: Straight-line embeddings of two rooted trees in the plane. Discrete & Computational Geometry 21(4), 603–613 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph and Combinatorics 17, 717–728 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. Sugiyama, K.: Graph Drawing and Applications. World Scientific, Singapore (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Di Giacomo, E., Liotta, G. (2010). The Hamiltonian Augmentation Problem and Its Applications to Graph Drawing. In: Rahman, M.S., Fujita, S. (eds) WALCOM: Algorithms and Computation. WALCOM 2010. Lecture Notes in Computer Science, vol 5942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11440-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-11440-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11439-7

  • Online ISBN: 978-3-642-11440-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics