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80
Ununfoldable Polyhedra
, 1999
"... A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can inde ..."
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Cited by 16 (9 self)
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A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.
On the Complexity of Optimization Problems for 3Dimensional Convex Polyhedra and Decision Trees
 Comput. Geom. Theory Appl
, 1995
"... We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent i ..."
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We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computeraided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3connected. Unfortunately, algorithmic problems dealing with 3dimensional convex polyhedra ...
A Constructive Enumeration of Fullerenes
"... In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regul ..."
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Cited by 16 (4 self)
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In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regular spherical maps with no face having more than 6 edges in its boundary.
Realization Spaces of 4Polytopes are Universal
 BULL. AMER. MATH. SOC
, 1995
"... Let P be a ddimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report ..."
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Cited by 16 (4 self)
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Let P be a ddimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report
Enumeration and random realization of triangulated surfaces.arXiv:math.CO
"... We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1 ..."
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Cited by 15 (9 self)
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We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 14 (0 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
Rigidity of polyhedral surfaces
, 2006
"... We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational a ..."
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Cited by 13 (5 self)
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We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include energies used by Colin de Verdiere, Braegger, Rivin, CohenKenyonPropp, Leibon and BobenkoSpringborn for variational principles on triangulated surfaces. Our study is based on a set of identities satisfied by the derivative of the cosine law. These identities which exhibit similarity in all spaces of constant curvature are probably a discrete analogous of the Bianchi identity.
Variational Principles for Circle Patterns
, 2003
"... A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given interse ..."
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Cited by 13 (4 self)
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A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow conelike singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The critical points correspond to solutions of the circle pattern problem. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz’ theorem on the geometric realizability of abstract polyhedra. We derive the
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. ..."
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Cited by 12 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.