# A Tight Bound for β-Skeleton of Minimum Weight Triangulations

## Abstract

In this paper, we prove a tight bound for *β* value \(
\left( {\beta = \frac{{\sqrt {2\sqrt 3 + 9} }}
{3}} \right)
\)
) such that being less than this value,the *β*-skeleton of a planar point set may not belong to the minimum weight triangulation of this set, while being equal to or greater than this value, the *β*-skeleton always belongs to the minimum weight triangulation. Thus, we settled the conjecture of the tight bound for *β*-skeleton of minimum weight triangulation by Mark Keil. We also present a new sufficient condition for identifying a subgraph of minimum weight triangulation of a planar *n*-point set. The identified subgraph could be different from all the known subgraphs, and the subgraph can be found in *O*(*n* ^{2} *log n*) time.

## Keywords

Line Segment Convex Hull Computational Geometry Internal Edge Primal Plane## Preview

Unable to display preview. Download preview PDF.

## References

- [AC93]E. Anagnostou and D. Corneil, Polynomial time instances of the minimum weight triangulation problem, Computational Geometry: Theory and applications, vol. 3, pp. 247–259, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
- [CGT95]S.-W. Cheng, M. Golin and J. Tsang, Expected case analysis of b-skeletons with applications to the construction of minimum weight triangulations, CCCG Conference Proceedings, P.Q., Canada, pp. 279–284, 1995.Google Scholar
- [BDE96]P. Bose, L. Devroye, and W. Evens, Diamonds are not a minimum weight triangulation’s best friend,
*Proceedings of 8th CCCG*, 1996, Ottawa, pp. 68–73.Google Scholar - [CX96]S.-W. Cheng and Y.-F. Xu, Approaching the largest β-skeleton within the minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996.Google Scholar
- [DK98]Y. Dai and N. Katoh, On computing new classes of optimal triangulations with angular constraints, Proceedings on 4th annual international conference of Computing and Combinatorics, LNCS 1449, pp.15–24.Google Scholar
- [DM96]M. T. Dickerson, M. H. Montague, The exact minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996.Google Scholar
- [Gi79]P. D. Gilbert, New results in planar triangulations, TR-850, University of Illinois Coordinated science Lab, 1979.Google Scholar
- [GJ79]M. Garey and D. Johnson, Computer and Intractability. A guide to the theory of NP-completeness, Freeman, 1979.Google Scholar
- [Ke94]J. M. Keil, Computing a subgraph of the minimum weight triangulation, Computational Geometry: Theory and Applications pp. 13–26, 4 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
- [KR85]D. Kirkpatrick and J. Radke, A framework for computational morphology, in G. Toussaint, ed., Computational Geometry, Elsevier, 1985, pp. 217–248.Google Scholar
- [Kl80]G. Klinesek, Minimal triangulations of polygonal domains, Ann. Discrete Math., pp. 121–123, 9 (1980).MathSciNetCrossRefGoogle Scholar
- [MR92]H. Meijer and D. Rappaport, Computing the minimum weight triangulation of a set of linearly ordered points, Information Processing Letters, vol. 42, pp. 35–38, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
- [O’R93]J. O’Rourke, Computational Geometry In C, Cambridge University Press, 1993.Google Scholar
- [MWX96]A. Mirzain, C. Wang and Y. Xu, On stable line segments in triangulations, Proceedings of 8th CCCG, Ottawa, 1996, pp.68–73.Google Scholar
- [WX96]C. Wang and Y. Xu, Minimum weight triangulations with convex layers constraint, to appear J. of Global Optimization.Google Scholar
- [WCX97]C. Wang, F. Chin, and Y. Xu, A new subgraph of Minimum weight triangulations, J. of Combinational Optimization Vol 1, No. 2, pp. 115–127.Google Scholar
- [XZ96]Y. Xu, D. Zhou, Improved heuristics for the minimum weight triangulation, Acta Mathematics Applicatae Sinica, vol. 11, no. 4, pp. 359–368, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
- [Yan95]B. Yang, A better subgraph of the minimum weight triangulation, The
**IPL**, Vol.56, pp. 255–258.Google Scholar - [YXY94]B. Yang, Y. Xu and Z. You, A chain decomposition algorithm for the proof of a property on minimum weight triangulations, Proc. 5th International Symposium on Algorithms and Computation (ISAAC’94), LNCS 834, Springer-Verlag, pp. 423–427, 1994.Google Scholar