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Finding Hamiltonian Cycles in Delaunay Triangulations Is NPComplete
 IN PROC. 4TH CANAD. CONF. COMPUT. GEOM
, 1994
"... It is shown that it is an NPcomplete problem to determine whether a Delaunay triangulation or an inscribable polyhedron has a Hamiltonian cycle. It is also shown that there exist nondegenerate Delaunay triangulations and simplicial, inscribable polyhedra without 2factors. ..."
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Cited by 18 (1 self)
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It is shown that it is an NPcomplete problem to determine whether a Delaunay triangulation or an inscribable polyhedron has a Hamiltonian cycle. It is also shown that there exist nondegenerate Delaunay triangulations and simplicial, inscribable polyhedra without 2factors.
GraphTheoretical Conditions for Inscribability and Delaunay Realizability
 Proceedings of the 6th Canadian Conference on Computational Geometry
, 1995
"... We present new graphtheoretical conditions for polyhedra of inscribable type and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is of inscribable type. These resu ..."
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Cited by 16 (3 self)
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We present new graphtheoretical conditions for polyhedra of inscribable type and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is of inscribable type. These results have several consequences: ffl All 4connected polyhedra are of inscribable type. ffl All simplicial polyhedra in which all vertex degrees are between 4 and 6, inclusive, are of inscribable type. ffl All triangulations without chords or nonfacial triangles are realizable as combinatorially equivalent Delaunay triangulations. We also strengthen some earlier results about matchings in polyhedra of inscribable type. Specifically, we show that any nonbipartite polyhedron of inscribable type has a perfect matching containing any specified edge, and that any bipartite polyhedron of inscribable type has a perfect matching containing any two specified disjoint edges. We give examples showing t...
A LinearTime Algorithm For Testing The Inscribability Of Trivalent Polyhedra
, 1995
"... We present an algorithm for testing the inscribability of a trivalent polyhedron or, equivalently, testing the circumscribability of a simplicial polyhedron. Our algorithm, which runs in linear time and uses only lowprecision integer arithmetic, is based on a purely combinatorial characterization o ..."
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Cited by 13 (4 self)
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We present an algorithm for testing the inscribability of a trivalent polyhedron or, equivalently, testing the circumscribability of a simplicial polyhedron. Our algorithm, which runs in linear time and uses only lowprecision integer arithmetic, is based on a purely combinatorial characterization of inscribable trivalent polyhedra. 1. Introduction A polyhedron is inscribable if it is combinatorially equivalent to the edges and vertices of the convex hull of a set of noncoplanar points on the surface of the sphere. The question of characterizing the combinatorial structure of inscribable polyhedra dates back to Descartes 1 , and was formally posed as an open problem by Jakob Steiner in 1832 2 . A history of the problem and some related questions can be found in (Refs. 3,4,5). Recent interest in the problem has been stimulated by the close connection between inscribable polyhedra and Delaunay triangulations 6 , and the importance of Delaunay triangulations as a fundamental struc...
Convex polyhedra in Lorentzian spaceforms
 Asian J. of Math
, 2001
"... Abstract. Aleksandrov [Ale51] characterized the metrics induced on convex polyhedra in E3, H3 and S3. We give analogs for compact and complete polyhedra in Lorentzian spaceforms. There are three types of convex polyhedra in the de Sitter space S3 1. One, which includes generalized hyperbolic polyhe ..."
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Cited by 13 (8 self)
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Abstract. Aleksandrov [Ale51] characterized the metrics induced on convex polyhedra in E3, H3 and S3. We give analogs for compact and complete polyhedra in Lorentzian spaceforms. There are three types of convex polyhedra in the de Sitter space S3 1. One, which includes generalized hyperbolic polyhedra, was treated in [Sch98a]. For the second, we characterize the induced metrics, and show that each is obtained on a unique polyhedron satisfying a natural condition at infinity. For the last type — compact polyhedra bounding compact domains — we describe the induced metrics, and give an existence and uniqueness result for a smaller class of metrics. The results on complete polyhedra are consequences of the study of the metrics induced on convex polyhedra in a natural extension of H3 by S3 1. We also characterize the metrics induced on compact, convex polyhedra in the Minkowski space E3 1. Those description are partly similar to those obtained in the Riemannian cases, but they also involve new elements of a metric and combinatorial nature. Résumé. Aleksandrov [Ale51] a caractérisé les métriques induites sur les polyèdres convexes dans E3, H3 et S3. On donne des résultats similaires pour les polyèdres compacts ou complets dans les formes d’espace lorentziennes. On distingue trois types de polyèdres convexes dans l’espace de Sitter S3 1. L’un d’eux, incluant
Circle packings of maps in polynomial time
 Eur. J. Comb
, 1997
"... The AndreevKoebeThurston circle packing theorem is generalized and improved in two ways. First, we get simultaneous circle packings of the map and its dual map so that, in the corresponding straightline representations of the map and the dual, any two edges dual to each other are perpendicular. N ..."
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Cited by 9 (2 self)
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The AndreevKoebeThurston circle packing theorem is generalized and improved in two ways. First, we get simultaneous circle packings of the map and its dual map so that, in the corresponding straightline representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primaldual circle packing representation in a surface of constant curvature is that its universal cover is 3connected (the map has no “planar” 2separations). Secondly, an algorithm is obtained that given a map M and a rational number ε>0 finds an εapproximation for the radii and the coordinates of the centres for the primaldual circle packing representation of M. The algorithm is polynomial in E(M)  and log(1/ε). In particular, for a map without planar 2separations on an arbitrary surface we have a polynomial time algorithm for simultaneous geodesic convex representations of the map and its dual so that only edges dual to each other cross, and the angles at the crossings are arbitrarily close to π
Polyhedra of Small Order and Their Hamiltonian Properties
, 1995
"... ... The results of the enumeration were used to systematically search for certain smallest nonHamiltonian polyhedral graphs. In particular, the smallest nonHamiltonian planar graphs satisfying certain toughnesslike properties are presented here, as are the smallest nonHamiltonian, 3connected, D ..."
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Cited by 8 (1 self)
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... The results of the enumeration were used to systematically search for certain smallest nonHamiltonian polyhedral graphs. In particular, the smallest nonHamiltonian planar graphs satisfying certain toughnesslike properties are presented here, as are the smallest nonHamiltonian, 3connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest nonHamiltonian, inscribable polyhedra are also given.
Connections between thetagraphs, Delaunay triangulations, and orthogonal surfaces
 In Proceedings of the 36th International Conference on Graph Theoretic Concepts in Computer Science (WG 2010
, 2010
"... Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulard ..."
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Cited by 5 (2 self)
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Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulardistance Delaunay triangulations, introduced by Chew, have been shown to be plane 2spanners of the 2D Euclidean complete graph, i.e., the distance in the TDDelaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6graph defined in the 2D Euclidean space, namely the halfΘ6graph, composed of the evencone edges of the Θ6graph. Our main contribution is to show that these graphs are exactly the TDDelaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish: – Every Θ6graph is the union of two spanning TDDelaunay graphs. In particular, Θ6graphs are 2spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6graphs are tspanners for some constant t, and Θ7graphs were only known to be tspanners for t ≈ 7.562. – Every plane triangulation is TDDelaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TDDelaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.
A Conformal Energy for Simplicial Surfaces
 COMBINATORIAL AND COMPUTATIONAL GEOMETRY
, 2005
"... A new functional for simplicial surfaces is suggested. It is invariant with respect to Möbius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. As an application a bending energy for discrete thinshells is derived. ..."
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Cited by 4 (1 self)
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A new functional for simplicial surfaces is suggested. It is invariant with respect to Möbius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. As an application a bending energy for discrete thinshells is derived.