Graphs and Combinatorics

, Volume 31, Issue 1, pp 35–57 | Cite as

Parity-Constrained Triangulations with Steiner Points

Original Paper
  • 108 Downloads

Abstract

Let \({P\subset\mathbb{R}^{2}}\) be a set of n points, of which k lie in the interior of the convex hull CH(P) of P. Let us call a triangulation T of P even (odd) if and only if all its vertices have even (odd) degree, and pseudo-even (pseudo-odd) if at least the k interior vertices have even (odd) degree. On the one hand, triangulations having all its interior vertices of even degree have one nice property; their vertices can be 3-colored, see (Heawood in Quart J Pure Math 29:270–285, 1898, Steinberg in A source book for challenges and directions, vol 55. Elsevier, Amsterdam, pp 211–248, 1993, Diks et al. in Lecture notes in computer science, vol 2573. Springer, Berlin, pp 138–149, 2002). On the other hand, odd triangulations have recently found an application in the colored version of the classic “Happy Ending Problem” of Erdős and Szekeres, see (Aichholzer et al. in SIAM J Discrete Math 23(4):2147–2155, 2010). It is easy to prove that there are sets of points that admit neither pseudo-even nor pseudo-odd triangulations. In this paper we show nonetheless how to construct a set of Steiner points SS(P) of size at most \({\frac{k}{3} + c}\) , where c is a positive constant, such that a pseudo-even (pseudo-odd) triangulation can be constructed on \({P \cup S}\) . Moreover, we also show that even (odd) triangulations can always be constructed using at most \({\frac{n}{3} + c}\) Steiner points, where again c is a positive constant. Our constructions have the property that all but at most two Steiner points lie in the interior of CH(P).

Keywords

Geometric Graphs Algorithmic Combinatorial Geometry Steinerpoints Triangulations Parity-constrained 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aichholzer, O., Hackl, T., Hoffmann, M., Pilz, A., Rote, G., Speckmann, B., Vogtenhuber, B.: Plane graphs with parity constraints. In: F.K.H.A., Dehne, M.L., Gavrilova, J.R., Sack, C.D., Tóth (eds.) WADS, Lecture Notes in Computer Science, vol. 5664, pp. 13–24. Springer, Berlin (2009)Google Scholar
  2. 2.
    Aichholzer, O., Hackl, T., Hoffmann, M., Pilz, A., Rote, G., Speckmann, B., Vogtenhuber, B.: Plane graphs with parity constraints. Graphs and Combinatorics, pp. 1–23 (2012). doi:10.1007/s00373-012-1247-y
  3. 3.
    Aichholzer O., Hackl T., Huemer C., Hurtado F., Vogtenhuber B.: Large bichromatic point sets admit empty monochromatic 4-gons. SIAM J. Discrete Math. 23(4), 2147–2155 (2010)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Alvarez, V.: Even triangulation of planar set of points with steiner points. In: EuroCG (2010)Google Scholar
  5. 5.
    Dey T.K., Dillencourt M.B., Ghosh S.K., Cahill J.M.: Triangulating with high connectivity. Comput. Geom. 8, 39–56 (1997)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Diks, K., Kowalik, L., Kurowski, M.: A new 3-color criterion for planar graphs. In: Kucera, L. (ed.) WG, Lecture Notes in Computer Science, vol. 2573, pp. 138–149. Springer, Berlin (2002)Google Scholar
  7. 7.
    Fisk S.: A short proof of Chvátal’s watchman theorem. J. Comb. Theory Ser. B 24(3), 374 (1978)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fleischner, H.: Gedanken zur Vier-Farben-Vermutung. Monatshefte für Mathematik 79(3), 201–211 (1975). doi:10.1007/BF01304073
  9. 9.
    Heawood P.J.: On the four-color map theorem. Q. J. Pure Math. 29, 270–285 (1898)MATHGoogle Scholar
  10. 10.
    Kooshesh, A.A., Moret, B.M.E.: Folding a triangulated simple polygon: structural and algorithmic results. In: F.K.H.A., Dehne, F., Fiala, W.W., Koczkodaj (eds.) ICCI, Lecture Notes in Computer Science, vol. 497, pp. 102–110. Springer, Berlin (1991)Google Scholar
  11. 11.
    Pilz, A.: Parity properties of geometric graphs. Master’s thesis, Graz University of Technology (2009)Google Scholar
  12. 12.
    Steinberg, R.: The state of the three color problem. In: J.W.K., John Gimbel, L.V., Quintas (eds.) Quo Vadis, Graph Theory? A Source Book for Challenges and Directions. Annals of Discrete Mathematics, vol. 55, pp. 211–248. Elsevier, Amsterdam (1993). doi:10.1016/S0167-5060(08)70391-1. URL http://www.sciencedirect.com/science/article/pii/S0167506008703911
  13. 13.
    Urrutia, J., Peláez, C., Ramírez-Vigueras, A.: Triangulations with many points of even degree. In: CCCG, pp. 103–106 (2010)Google Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Information Systems GroupUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations