Document Open Access Logo

Arc Diagrams, Flip Distances, and Hamiltonian Triangulations

Authors Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, Manuel Wettstein



PDF
Thumbnail PDF

File

LIPIcs.STACS.2015.197.pdf
  • Filesize: 0.67 MB
  • 14 pages

Document Identifiers

Author Details

Jean Cardinal
Michael Hoffmann
Vincent Kusters
Csaba D. Tóth
Manuel Wettstein

Cite AsGet BibTex

Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc Diagrams, Flip Distances, and Hamiltonian Triangulations. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 197-210, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.STACS.2015.197

Abstract

We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.
Keywords
  • graph embeddings
  • edge flips
  • flip graph
  • separating triangles

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, and Gelasio Salazar. Shellable drawings and the cylindrical crossing number of K_n. Discrete Comput. Geom., 52(4):743-753, 2014. Google Scholar
  2. Oswin Aichholzer, Clemens Huemer, and Hannes Krasser. Triangulations without pointed spanning trees. Comput. Geom. Theory Appl., 40(1):79-83, 2008. Google Scholar
  3. T. Alastair and J. Nicholson. Permutation procedure for minimising the number of crossings in a network. Proc. IEE, 115(1):21-26, 1968. Google Scholar
  4. Patrizio Angelini, David Eppstein, Fabrizio Frati, Michael Kaufmann, Sylvain Lazard, Tamara Mchedlidze, Monique Teillaud, and Alexander Wolff. Universal point sets for drawing planar graphs with circular arcs. Journal of Graph Algorithms and Applications, 18(3):313-324, 2014. Google Scholar
  5. Frank Bernhart and Paul C. Kainen. The book thickness of a graph. J. Combin. Theory, Ser. B 27:320-331, 1979. Google Scholar
  6. J. Blažek and M. Koman. A minimal problem concerning complete plane graphs. In M. Fiedler, editor, Theory of graphs and its applications, pages 113-117. Czech. Acad. of Sci., 1964. Google Scholar
  7. John Adrian Bondy and U. S. R. Murty. Graph Theory, volume 244 of Graduate texts in Mathematics. Springer, Berlin, 2008. Google Scholar
  8. Prosenjit Bose, Jurek Czyzowicz, Zhicheng Gao, Pat Morin, and David R. Wood. Simultaneous diagonal flips in plane triangulations. J. Graph Theory, 54(4):307-330, 2007. Google Scholar
  9. Prosenjit Bose and Ferran Hurtado. Flips in planar graphs. Comput. Geom. Theory Appl., 42(1):60-80, 2009. Google Scholar
  10. Prosenjit Bose, Dana Jansens, André van Renssen, Maria Saumell, and Sander Verdonschot. Making triangulations 4-connected using flips. Comput. Geom. Theory Appl., 47(2):187-197, 2014. Google Scholar
  11. Prosenjit Bose and Sander Verdonschot. A history of flips in combinatorial triangulations. In Computational Geometry - XIV Spanish Meeting on Computational Geometry, EGC 2011, volume 7579 of LNCS, pages 29-44. Springer, Berlin, 2012. Google Scholar
  12. Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer, Berlin, 4 edition, 2010. Google Scholar
  13. David Eppstein. Universal point sets for planar graph drawings with circular arcs. Presentation at the 25th Canadian Conference on Computational Geometry, Waterloo, Canada, http://www.ics.uci.edu/~eppstein/pubs/AngEppFra-CCCG-13-slides.pdf, 2013.
  14. Emilio Di Giacomo, Walter Didimo, and Giuseppe Liotta. Spine and radial drawings. In Roberto Tamassia, editor, Handbook of Graph Drawing and Visualization, chapter 8, pages 247-284. CRC press, Boca Raton, FL, 2013. Google Scholar
  15. Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Stephen K. Wismath. Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl., 30(1):1-23, 2005. Google Scholar
  16. Branko Grünbaum. Convex Polytopes, volume 221 of Graduate texts in Mathematics. Springer, Berlin, 2003. Google Scholar
  17. Jochen Harant, Mirko Horňák, and Zdislaw Skupień. Separating 3-cycles in plane triangulations. Discrete Math., 239(1):127-136, 2001. Google Scholar
  18. Goos Kant. A more compact visibility representation. Int. J. Comput. Geometry Appl., 7(3):197-210, 1997. Google Scholar
  19. Michael Kaufmann and Roland Wiese. Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl., 6(1):115-129, 2002. Google Scholar
  20. Hideo Komuro. The diagonal flips of triangulations on the sphere. Yokohama Math. J., 44:115-122, 1997. Google Scholar
  21. Ryuichi Mori, Atsuhiro Nakamoto, and Katsuhiro Ota. Diagonal flips in Hamiltonian triangulations on the sphere. Graphs Combin., 19(3):413-418, 2003. Google Scholar
  22. János Pach and Rephael Wenger. Embedding planar graphs at fixed vertex locations. Graphs Combin., 17:717-728, 2001. Google Scholar
  23. Lionel Pournin. The diameter of associahedra. Adv. Math., 259:13-42, 2014. Google Scholar
  24. Thomas L. Saaty. The minimum number of intersections in complete graphs. Proceedings of the National Academy of Sciences of the United States of America, 52:688-690, 1964. Google Scholar
  25. Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc., 1:647-682, 1988. Google Scholar
  26. Avi Wigderson. The complexity of the Hamiltonian circuit problem for maximal planar graphs. Technical Report 298, Princeton University, 1982. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail