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125
Circle Patterns With The Combinatorics Of The Square Grid
 Duke Math. J
, 1997
"... . Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is infinite dimensional. In Particular, Doyle's conjecture is false in this setting. Mobius inv ..."
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Cited by 43 (1 self)
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. Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is infinite dimensional. In Particular, Doyle's conjecture is false in this setting. Mobius invariants of circle patterns are introduced, and turn out to be discrete analogs of the Schwarzian derivative. The invariants satisfy a nonlinear discrete version of the CauchyRiemann equations. A global analysis of the solutions of these equations yields a rigidity theorem characterizing the Doyle spirals. It is also shown that by prescribing boundary values for the Mobius invariants, and solving the appropriate Dirichlet problem, a locally univalent meromorphic function can be approximated by circle patterns. 1991 Mathematics Subject Classification. 30C99, 05B40, 30D30, 31A05, 31C20, 30G25. Key words and phrases. Meromorphic functions, Schwarzian derivative, rigidity, error function, Dirichlet ...
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 33 (8 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...
GMap: Visualizing Graphs and Clusters as Maps
, 2009
"... Information visualization is essential in making sense out of large data sets. Often, highdimensional data are visualized as a collection of points in 2dimensional space through dimensionality reduction techniques. However, these traditional methods often do not capture well the underlying structu ..."
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Cited by 32 (21 self)
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Information visualization is essential in making sense out of large data sets. Often, highdimensional data are visualized as a collection of points in 2dimensional space through dimensionality reduction techniques. However, these traditional methods often do not capture well the underlying structural information, clustering, and neighborhoods. In this paper, we describe GMap, a practical tool for visualizing relational data with geographiclike maps. We illustrate the effectiveness of this approach with examples from several domains. All the maps referenced in this paper can be found in www.research.att.com/˜yifanhu/GMap.
Circle packing: a mathematical tale
 Notices Amer. Math. Soc
, 2003
"... The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creat ..."
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Cited by 27 (2 self)
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The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere ” packing: rather than being fixed in size, our circles must adjust their radii in tightly choreographed ways if they hope to fit together in a specified pattern. In posing this as a mathematical tale, I am asking the reader for some latitude. From a tale you expect truth without all the details; you know that the storyteller will be playing with the plot and timing; you let pictures carry part of the story. We all hope for deep insights, but perhaps sometimes a simple story with a few new twists is enough—may you enjoy this tale in that spirit. Readers who wish to dig into the details can consult the “Reader’s Guide ” at the end. Once Upon a Time … From wagon wheel to mythical symbol, predating history, perfect form to ancient geometers, companion to π, the circle is perhaps the most celebrated object in mathematics.
Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs
, 2002
"... We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running t ..."
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Cited by 24 (2 self)
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We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running time bounded by an exponential with a sublinear exponent. For precision disk graphs of bounded radius ratio, we show that the problem is xed parameter tractable with respect to parameter k.
The Colin de Verdière graph parameter
, 1997
"... In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless emb ..."
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Cited by 24 (3 self)
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In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.
Hexagonal circle patterns and integrable systems: Patterns with constant angles
, 2003
"... Hexagonal circle patterns with constant intersection angles are introduced and studied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs of holomorphic mappings z c and log z are constructed as spec ..."
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Cited by 23 (2 self)
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Hexagonal circle patterns with constant intersection angles are introduced and studied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs of holomorphic mappings z c and log z are constructed as special isomonodromic solutions. Circle patterns studied in the paper include Schramm’s circle patterns with the combinatorics of the square grid as a special case.
Rectangular Layouts and Contact Graphs
, 2007
"... Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the areaoptimization problem and show that it is NPhard t ..."
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Cited by 23 (3 self)
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Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the areaoptimization problem and show that it is NPhard to find a minimumarea rectangular layout of a given contact graph. We present O(n)time algorithms that construct O(n2)area rectangular layouts for general contact graphs and O(n log n)area rectangular layouts for trees. (For trees, this is an O(log n)approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require Ω(n2) (rsp., Ω(n log n)) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of rectangular duals. A corollary to our results relates the class of graphs that admit rectangular layouts to rectangle of influence drawings.
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 21 (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
Representing Graphs by Disks and Balls (a survey of recognitioncomplexity results)
"... Practical applications, like radio frequency assignments, led to the definition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recogniti ..."
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Cited by 20 (1 self)
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Practical applications, like radio frequency assignments, led to the definition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recognitioncomplexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about boundedratio disk representations. We prove that the recognition of diskintersection graphs (in the unbounded ratio case) is NPhard. This result is proved in a more general setting of noncrossing arcconnected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unitball contact graphs is NPhard in dimensions 3, 4, and 8 (24).