# On the relationships among constrained geometric structures

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## Abstract

In this paper, we show the inclusion relation among several constrained geometric structures. In particular, we examine the constrained relative neighborhood graph in relation with other constrained geometric structures such as the constrained minimum spanning tree, constrained Gabriel graph, straight-line dual of bounded Voronoi diagram and the constrained Delaunay triangulation. We modify a linear time algorithm for computing the relative neighborhood graph from the Delaunay triangulation and show that the constrained relative neighborhood graph can also be computed in linear-time from the constrained Delaunay triangulation in the (*ℜ*^{2}, *L*_{ p }) metric space.

## Keywords

Voronoi Diagram Directed Edge Delaunay Triangulation Inclusion Relation Constrain Delaunay Triangulation
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## References

- [1]F. Aurenhammer, “
*Voronoi Diagrams — A Survey of a Fundamental Geometric Data Structure*”, ACM Comp. Surveys, Vol. 23, No. 3, Sep. 1991, pp. 345–406.CrossRefGoogle Scholar - [2]L. P. Chew, “
*Constrained Delaunay Triangulations*”, Algorithmica**4**(1989), pp. 97–108.CrossRefGoogle Scholar - [3]H. N. Gabow and R. E. Tarjan, “
*A Linear-Time Algorithm for a Special Case of Disjoint Set Union*”, Journal of Computer and System Sciences 30, 1985, pp. 209–221.Google Scholar - [4]K. R. Gabriel and R. R. Sokal, “
*A New Statistical Approach to Geographic Variation Analysis*”, Systematic Zoology 18 (1969), pp. 259–278.Google Scholar - [5]J. W. Jaromczyk and M. Kowaluk, “
*A Note on Relative Neighborhood Graphs*”, Proceedings of the 3^{rd}Annual Symposium on Computational Geometry, Waterloo, Canada, 1987, pp. 233–241.Google Scholar - [6]J. W. Jaromczyk, M. Kowaluk and F. F. Yao, “
*An Optimal Algorithm for Constructing Β-skeletons in L*_{p}*Metric*”, (manuscript, 1992) to appear in the SIAM Journal of Computing.Google Scholar - [7]D. T. Lee and A. K. Lin, “
*Generalized Delaunay Triangulation for Planar Graphs*”, Discrete and Computational Geometry, 1 (1986), pp. 201–217.Google Scholar - [8]A. Lingas, “
*Voronoi Diagrams with Barriers and the Shortest Diagonal Problem*”, Information Processing Letters 32 (1989), pp. 191–198.Google Scholar - [9]A. Lingas, “
*A Simple Linear-Time Algorithm for Constructing the Relative Neighborhood Graph from the Delaunay Triangulation*”, 1992, (submitted).Google Scholar - [10]F. P. Preparata and M. I. Shamos, “
*Computational Geometry, An Introduction*”, Springer-Verlag, 1985.Google Scholar - [11]R. Seidel, “
*Constrained Delaunay Triangulations and Voronoi Diagrams with Obstacles*”, In Rep. 260, IIG-TU, Graz, Australia, 1988, pp. 178–191.Google Scholar - [12]T. H. Su and R. C. Chang, “
*Computing the Constrained Relative Neighborhood Graphs and Constrained Gabriel Graphs in Euclidean Plane*”, Pattern Recognition, Vol. 24, No. 3, 1991, pp. 221–230.CrossRefGoogle Scholar - [13]K. J. Supowit, “
*The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees*”, J. of Assoc. for Comput. Mach. Vol. 30, No. 3, July 1983, pp. 428–448.Google Scholar - [14]G. Toussaint, “
*The Relative Neighborhood Graph of a Finite Planar Set*”, Pattern Recognition, Vol. 12 (1979), pp. 261–268.CrossRefGoogle Scholar - [15]C. A. Wang and L. Schubert, “
*An Optimal Algorithm for Constructing the Delaunay Triangulation of a Set of Line Segments*”, Proceedings of the 3^{rd}Annual Symposium on Computational Geometry, June 8–10, 1987, Waterloo, Ontario, Canada, pp. 223–232.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1992