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30
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 43 (9 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
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 33 (13 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
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 32 (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 (8 self)
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We consider the problem of simultaneous embedding of planar graphs. There are two variants
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 (7 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 17 (2 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 16 (4 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 14 (9 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.
Rectangle and Square Representations of Planar Graphs
"... In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the recta ..."
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Cited by 10 (5 self)
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In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part we ask for representations by a dissections of a rectangle into squares. We