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26
Minimal surfaces from circle patterns: geometry from combinatorics
 Ann. of Math
"... The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studie ..."
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Cited by 48 (10 self)
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The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 33 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
On Simultaneous Planar Graph Embeddings
 COMPUT. GEOM
, 2003
"... We consider the problem of simultaneous embedding of planar graphs. There are two variants ..."
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Cited by 29 (9 self)
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We consider the problem of simultaneous embedding of planar graphs. There are two variants
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Hyperbolic And Parabolic Packings
 Discrete Comput. Geom
, 1994
"... . The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the 1skeleton of a triangulation of an open disk. G is said to be CP parabolic [respectively CP hyperbolic], if there is a locally finite disk packing P in the plane [res ..."
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Cited by 22 (6 self)
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. The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the 1skeleton of a triangulation of an open disk. G is said to be CP parabolic [respectively CP hyperbolic], if there is a locally finite disk packing P in the plane [respectively, the unit disk] with contacts graph G . Several criteria for deciding whether G is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial criterion. A criterion in terms of the random walk says that if the random walk on G is recurrent, then G is CP parabolic. Conversely, if G has bounded valence and the random walk on G is transient, then G is CP hyperbolic. We shall also give a new proof that G is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that if G is CP hyperbolic and D is any simply connected proper subdomain of the plane...
On the cover time of planar graphs
 Electron. Comm. Probab
"... The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any nvertex, connected graph is at least ( 1 + o(1) ) n log n and at most ( 1 + o(1) ) 4 27 n3. This paper proves tha ..."
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Cited by 15 (1 self)
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The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any nvertex, connected graph is at least ( 1 + o(1) ) n log n and at most ( 1 + o(1) ) 4 27 n3. This paper proves that for boundeddegree planar graphs the cover time is at least cn(log n) 2, and at most 6n 2, where c is a positive constant depending only on the maximal degree of the graph. 1
Optimal Möbius Transformations for Information Visualization and Meshing
 Meshing, WADS 2001, Lecture Notes in Computer Science 2125
, 2001
"... . We give lineartime quasiconvex programming algorithms for ..."
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Cited by 15 (3 self)
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. We give lineartime quasiconvex programming algorithms for
Simultaneous embedding of a planar graph and its dual on the grid
, 2002
"... Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar ..."
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Cited by 13 (8 self)
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Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straightline segments and the only crossings are between primaldual pairs of edges. We provide a lineartime algorithm that simultaneously embeds a 3connected planar graph and its dual on a (2n −2) ×(2n −2) integer grid, where n is the total number of vertices in the graph and its dual. Furthermore our embedding algorithm satisfies the two natural requirements mentioned above.
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