Toughness and Delaunay triangulations

We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graphG is1-tough 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 1-tough, 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 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.

[1]  Michael B. Dillencourt,et al.  Realizability of Delaunay Triangulations , 1990, Inf. Process. Lett..

[2]  Kevin Q. Brown,et al.  Voronoi Diagrams from Convex Hulls , 1979, Inf. Process. Lett..

[3]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[4]  H. Whitney.  A Theorem on Graphs , 1931 .

[5]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[6]  H. Coxeter,et al.  Introduction to Geometry. , 1961 .

[7]  S. Louis Hakimi,et al.  Recognizing tough graphs is NP-hard , 1990, Discret. Appl. Math..

[8]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[9]  A. Lingus,et al.  The Greedy and Delaunay Triangulations are Not Bad in the Average Case , 1986, Inf. Process. Lett..

[10]  Takao Nishizeki.  A 1-tough nonhamiltonian maximal planar graph , 1980, Discret. Math..

[11]  Glenn K. Manacher,et al.  Neither the Greedy Nor the Delaunay Triangulation of a Planar Point Set Approximates the Optimal Triangulation , 1979, Inf. Process. Lett..

[12]  D. Barnette,et al.  Hamiltonian circuits on 3-polytopes , 1970 .

[13]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[14]  E. Bolker,et al.  Generalized Dirichlet tessellations , 1986 .

[15]  Michael B. Dillencourt,et al.  An upper bound on the shortness exponent of inscribable polytopes , 1989, J. Comb. Theory, Ser. B.

[16]  Branko Grünbaum,et al.  Some problems on polyhedra , 1987 .

[17]  O. Ore.  The Four-Color Problem , 1967 .

[18]  Vasek Chvátal,et al.  Tough graphs and hamiltonian circuits , 1973, Discret. Math..

[19]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1986, Discret. Comput. Geom..

[20]  David P. Dobkin,et al.  Delaunay graphs are almost as good as complete graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[21]  Vitit KANTABUTRA,et al.  Traveling Salesman Cycles are not Always Subgraphs of Voronoi Duals , 1983, Inf. Process. Lett..

[22]  Carl Gutwin,et al.  The Delauney Triangulation Closely Approximates the Complete Euclidean Graph , 1989, WADS.

[23]  Michael B. Dillencourt.  An upper bound on the shortness exponent of 1-tough, maximal planar graphs , 1991, Discret. Math..

[24]  Richard C. T. Lee,et al.  On the average length of Delaunay triangulations , 1984, BIT.

[25]  Joseph O'Rourke,et al.  The Computational Geometry Column #2 , 1987, COMG.

[26]  D. T. Lee,et al.  Computational Geometry—A Survey , 1984, IEEE Transactions on Computers.

[27]  Herbert Edelsbrunner,et al.  An acyclicity theorem for cell complexes ind dimension , 1990, Comb..

[28]  Herbert Edelsbrunner,et al.  An acyclicity theorem for cell complexes in d dimensions , 1989, SCG '89.

[29]  Ethan D. Bolker,et al.  Recognizing Dirichlet tessellations , 1985 .

[30]  David G. Kirkpatrick,et al.  A Note on Delaunay and Optimal Triangulations , 1980, Inf. Process. Lett..

[31]  Jean-Daniel Boissonnat,et al.  Geometric structures for three-dimensional shape representation , 1984, TOGS.

[32]  Michael B. Dillencourt.  Traveling Salesman Cycles are not Always Subgraphs of Delaunay Triangulations or of Minimum Weight Triangulations , 1987, Inf. Process. Lett..

[33]  J. O'Rourke,et al.  Connect-the-dots: a new heuristic , 1987 .

[34]  Errol L. Lloyd.  On triangulations of a set of points in the plane , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[35]  M. H. A. Newman,et al.  Topology . Elements of the topology of plane sets of points , 1939 .

[36]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[37]  Michael B. Dillencourt,et al.  A Non-Hamiltonian, Nondegenerate Delaunay Triangulation , 1987, Inf. Process. Lett..