Abstract
We state the following conjecture: any two planar n-point sets (that agree on the number of convex hull points) can be triangulated in a compatible manner, i.e., such that the resulting two planar graphs are isomorphic. The conjecture is proved true for point sets with at most three interior points. We further exhibit a class of point sets which can be triangulated compatibly with any other set (that satis?es the obvious size and hull restrictions). Finally, we prove that adding a small number of Steiner points (the number of interior points minus two) always allows for compatible triangulations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
O. Aichholzer, F. Aurenhammer, H. Krasser, Enumerating order types for small point sets with applications. 17th Ann. ACM Symp. Computational Geometry, Medford, MA, 2001 (to be presented).
B. Aronov, R. Seidel, D. Souvaine, On compatible triangulations of simple polygons. Computational Geometry: Theory and Applications 3 (1993), 27–35.
M. Bern, A. Sahai, Isomorphic triangulations and map conflaation. Personal communication, 2000.
J.E. Goodman, R. Pollack, Multidimensional sorting. SIAM J. Computing 12 (1983), 484–507.
E. Kranakis, J. Urrutia, Isomorphic triangulations with small number of Steiner points. Int’l J. Computaional Geometry & Applications 9 (1999), 171–180.
H. Krasser, Kompatible Triangulierungen ebener Punktmengen. Master Thesis, Institute for Theoretical Computer Science, Graz University of Technology, Graz, Austria, 1999.
A. Saalfeld, Joint triangulations and triangulation maps. Proc. 3rd Ann. ACM Sympos. Computational Geometry, Waterloo, Canada, 1987, 195–204.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aichholzer, O., Aurenhammer, F., Krasser, H., Hurtado, F. (2001). Towards Compatible Triangulations. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_12
Download citation
DOI: https://doi.org/10.1007/3-540-44679-6_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42494-9
Online ISBN: 978-3-540-44679-8
eBook Packages: Springer Book Archive