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Optimal Triangulation: Old and New Problems 2673. ConclusionWe give a list of unsolved problems related with optimal triangulations both from the theoreticaland the application aspects, and show some partial results on the problems in the paper.To keep our attention on the practical aspects of computing for computational geometry, somenew challenge problems on optimal triangulations need to be solved.References[1] O. Aichholzer, F. Aurenhammer and R. Hainz, New results on MW T subgraphs, Information ProcessingLetters, 69, 215-219,1999.[2] O. Aichholzer, D. Orden, F. Santos and B. Speckmannt, On the number of pseudo-triangulations of certainpoint sets, In Proc. CCCG’03, 141-144, Halifax, Nova Scotia, Canada, 2003.[3] E. Anagnostou and D. Corneil, Polynomial-time instances of the minimum weight triangulation problem,Computational Geometry: Theory and Application, 3,247-259, 1993.[4] F. Aurenhammer, Voronoi diagrams-a survey of a fundamental geometric data structure, ACM ComputingSurveys, 23, 345-405, 1991.[5] F. Aurenhammer and Y. F. Xu, Optimal triangulations, In PM. Pardalos and CA. Floudas, editor, Encyclopediaof Optimization, volume 4, 160-166, Kluwer Academic Publishing, 2000.[6] M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell and T. S. Tan, Edge insertion for optimal tiangulations,Dist. Comput. Geom, 10, 47-65, 1993.[7] M. Bern and D. Eppstein. Mesh generation and optimal triangulation, In : D. -Z. Du and F Hwang, editors,Computing in Euclidean Geometry, Lecture Notes Series in Computing 4, World Scientific, Singapore, 47-123, 1995.[8] M. Bern and D. Eppstein, Approximation algorithms for geometric problems, In D. S. Hochbaum, editors,Approximation Algorithms for NP-hard Problems, 290-345, PWS , 1995.[9] P. Bose, L. Devroye and W. Evans, Diamonds are not a minimum weight triangulation’s best friend, In Proc.CCCG’96, 68-73,1996.[10] S. W. Cheng, M. J. Golin and J. C. F. Tsang, Expected-case analysis of β-skeletons with applications to theconstruction of minimum-weight triangulations, In Proc. CCCG’95, 279-283, 1995.[11] S. W. Cheng, N. Katoh and M. Sugai, A study of the LMT-skeleton, in Proc. ISAAC’96, LNCS 1178, Springer-Verlag, 256-265, 1996.[12] D. Eppstein. Approximating the minimum weight triangulation, Disc. and Comp. Geometry, 11, 163-191, 1994.[13] S. Fortune, Voronoi Diagrams and Delaunay triangulations, in: D. -Z. Du and F. Hwang, ed., Computing inEuclidean Geometry, Lecture Notes Series in Computing 4, World Scientific, Singapore, 225-265, 1995.[14] P. D. Gilbert, New results in planar triangulation, report R-850, Coordinated Science Laboratory, University ofIllinois, 1979.[15] G. T. Klincsek, Minimal triangulations of polygonal domains, Ann. Discrete Math, 9, 127-128, 1980.[16] H. Meijer and D. Rappaport, Computing the minimum weight triangulation for a set of linearly orderedpoints, Information Processing Letters, 42, 35-38,1992.[17] A. Mirzain, C. A. Wang and Y. F. Xu, On stable line segmets in triangulations, in: Proc. CCCG’96, 68-73, 1996.[18] F. P. Preparata and M. I. Shamos, Computational Geometry: An Introuduction, Springer-Verlag, 1985.[19] G. Rote, Pseudotriangulations, polytopes, and how to expand linkages, In Proc of the 18th annual symposiumon Computational Geometry, 133-134, 2002.[20] Y. F. Xu, Minimum weight triangulation problem of a planar point set, Ph.D. Thesis. Institute of Applied Mathematics,Academia Sinica, Beijing, 1992.[21] Y. F. Xu, On stable line segments in all triangulations, Appl. Math. -JCU, 11B,2, 235-238, 1996.[22] Y. F. Xu, Minimum weight triangulations, In D. -Z. Du and P. M. Pardalos, editors, Handbook of CombinatorialOptimaization (Vol. 2), Kluwer Academic Publishers, 617-634, 1998.[23] Y.F.Xu, W.Q.Dai, N. Katoh and M. Ohsaki, Triangulating a convex polygon with small number of nonstandardbars, to appear in Proc. COCOON05, 2005.[24] Y. F. Xu and D. Zhou, Improved heuristics for the minimum weight triangulation, Acta Mathematicae ApplicataeSinica 4(11), 359-368, 1995.