Toughness and Delaunay triangulations
 Michael B. Dillencourt
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1toughness. A graphG is1tough if for any setP of vertices,c(G−P)≤G, wherec(G−P) is the number of components of the graph obtained by removingP and all attached edges fromG, and G is the number of vertices inG. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1tough, but not conversely. We also show that all Delaunay triangulationsT satisfy the following closely related property: for any setP of vertices the number of interior components ofT−P is at most P−2, where an interior component ofT−P is a component that contains no boundary vertex ofT. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.
 S. G. Akl. A note on Euclidean matchings, triangulations, and spanning trees.Journal of Combinatorics, Information and System Sciences,8(3):169–174, 1983.
 P. F. Ash and E. D. Bolker. Recognizing Dirichlet tessellations.Geometriae Dedicata,19(2):175–206, November 1985.
 P. F. Ash and E. D. Bolker. Generalized Dirichlet tesselations.Geometriae Dedicata,20(2):209–243, April 1986.
 C. A. Barefoot, R. Entringer, and H. Swart. Vulnerability in graphs—a comparative survey.Journal of Combinatorial Mathematics and Combinatorial Computing,1:13–22, 1987.
 D. Barnette and E. Jucovič. Hamiltonian circuits on 3polytopes.Journal of Combinatorial Theory,9(1):54–59, July 1970.
 D. Bauer, S. L. Hakimi, and E. Schmeichel. Recognizing tough graphs is NPhard.Discrete Mathematics, to appear.
 J. C. Bermond. Hamiltonian graphs. In L. W. Beineke and R. J. Wilson, editors,Selected Topics in Graph Theory, pp. 127–167. Academic Press, London, 1978.
 J.D. Boissonnat. Geometric structures for threedimensional storage representation.ACM Transactions on Graphics,3(4):256–286, October 1984.
 J.D. Boissonnat. Private Communication.
 J. A. Bondy and U. S. R. Murty.Graph Theory with Applications. NorthHolland, New York, 1976.
 K. Q. Brown. Voronoi diagrams from convex hulls.Information Processing Letters,9(5):223–228, December 1979.
 R. C. Chang and T. C. T. Lee. On the average length of Delaunay triangulations.BIT,24(3):269–273, 1984.
 V. Chvátal. Tough graphs and Hamiltonian circuits.Discrete Mathematics,5(3):215–228, July 1973.
 V. Chvátal. Hamiltonian cycles. In E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, editors,The Traveling Salesman Problem, pp. 403–429. Wiley, New York, 1985.
 H. S. M. Coxeter.Introduction to Geometry, second edition, Wiley, New York, 1969.
 R. W. Dawes and M. G. Rodrigues. Properties of 1tough graphs.Journal of Combinatorial Mathematics and Combinatorial Computing, to appear.
 L. De Floriani, B. Falcidieno, C. Pienovi, and G. Nagy. On sorting triangles in a Delaunay tessellation.Algorithmica, to appear.
 B. Delaunay. Sur la sphère vide.Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk,7(6):793–800, October 1934.
 M. B. Dillencourt. Traveling salesman cycles are not always subgraphs of Delaunay triangulations or of minimum weight triangulations.Information Processing Letters,24(5):339–342, March 1987.
 M. B. Dillencourt. A nonHamiltonian, nondegenerate Delaunay triangulation.Information Processing Letters,25(3):149–151, May 1987.
 M. B. Dillencourt. GraphTheoretical Properties of Algorithms Involving Delaunay Triangulations. Ph.D. thesis, University of Maryland, College Park, MD, April 1988. Also Computer Science Technical Report CSTR2059, University of Maryland, College Park, MD.
 M. B. Dillencourt. An upper bound on the shortness exponent of inscribable graphs.Journal of Combinatorial Theory, Series B,46(1):66–83, February 1989.
 M. B. Dillencourt. An upper bound on the shortness exponent of 1tough, maximal planar graphs.Discrete Mathematics, to appear.
 M. B. Dillencourt. Realizability of Delaunay triangulations.Information Processing Letters,33(6):283–287, February 1990.
 D. P. Dobkin, S. J. Friedman, and K. J. Supowit. Delaunay graphs are almost as good as complete graphs. InProceedings of the 28th IEEE Symposium on the Foundations of Computer Science, pp. 20–26, Los Angeles, CA, October 1987.
 H. Edelsbrunner,Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, Vol. 10. SpringerVerlag, Berlin, 1987.
 H. Edelsbrunner. An acyclicity theorem for cell complexes ind dimensions. InProceedings of the Fifth ACM Symposium on Computational Geometry, pp. 145–151, Saarbrücken, West Germany, June 1989.
 H. Edelsbrunner and R. Seidel. Voronoi diagrams and arrangements.Discrete and Computational Geometry,1(1):25–44, 1986.
 B. Grünbaum.Convex Polytopes. Wiley Interscience, New York, 1967.
 B. Grünbaum and G. C. Shephard. Some problems on polyhedra.Journal of Geometry,29(2):182–190, August 1987.
 V. Kantabutra. Traveling salesman cycles are not always subgraphs of Voronoi duals.Information Processing Letters,16(1):11–12, January 1983.
 J. M. Keil and C. A. Gutwin. The Delaunay triangulation closely approximates the complete Euclidean graph. InProceedings of the 1989 Workshop on Algorithms and Data Structures (WADS '89), pp. 47–56, Ottawa, August 1989. Lecture Notes in Computer Science, Vol. 382. SpringerVerlag, Berlin, 1989.
 D. G. Kirkpatrick. A note on Delaunay and optimal triangulations.Information Processing Letters,10(3):127–128, April 1980.
 D. T. Lee and F. P. Preparata. Computational geometry—a survey.IEEE Transactions on Computers,33(12):1072–1101, December 1984.
 A. Lingas. The greedy and Delaunay triangulations are not bad in the average case.Information Processing Letters,22(1):25–31, January 1986.
 E. L. Lloyd. On triangulations of a set of points in the plane. InProceedings of the Eighteenth Annual IEEE Symposium on the Foundations of Computing, pp. 228–240, Providence, RI, October 1977.
 G. K. Manacher and A. L. Zobrist. Neither the greedy nor the Delaunay triangulation of a planar point set approximates the optimal triangulation.Information Processing Letters,9(1):31–34, July 1979.
 C. Mathieu. Some problems in computational geometry.Algorithmica,2(1):131–133, 1987.
 M. H. A. Newman.Elements of the Topology of Plane Sets of Points, second edition. Cambridge University Press, Cambridge, 1951.
 T. Nishizeki. A 1tough Nonhamiltonian maximal planar graph.Discrete Mathematics,30(3):305–307, June 1980.
 O. Ore,The FourColor Problem. Academic Press, New York, 1967.
 J. O'Rourke. The computational geometry column.Computer Graphics,20(5):232–234, October 1986. Also inSIGACT News,18(1):17–19, Summer 1986.
 J. O'Rourke, H. Booth, and R. Washington. Connectthedots: A new heuristic.Computer Vision, Graphics, and Image Processing,39(2):258–266, August 1987.
 F. P. Preparata and M. I. Shamos.Computational Geometry: An Introduction. SpringerVerlag, New York, 1985.
 M. I. Shamos. Computational Geometry. Ph.D. thesis, Yale University, New Haven, CT, 1978.
 W. T. Tutte. The factorization of linear graphs.Journal of the London Mathematical Society,22:107–111, 1947.
 H. Whitney. A theorem on graphs.Annals of Mathematics,32:378–390, 1931.
 Title
 Toughness and Delaunay triangulations
 Journal

Discrete & Computational Geometry
Volume 5, Issue 1 , pp 575601
 Cover Date
 19901201
 DOI
 10.1007/BF02187810
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Michael B. Dillencourt ^{(1)}
 Author Affiliations

 1. Center for Automation Research and Institute of Advanced Computer Studies, University of Maryland, 20742, College Park, MD, USA