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31
A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows ..."
Abstract

Cited by 520 (40 self)
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Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
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 invarian ..."
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Cited by 33 (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 ...
Arc triangulations
 PROC. 26TH EUR. WORKSH. COMP. GEOMETRY (EUROCG’10)
, 2010
"... The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alter ..."
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Cited by 27 (2 self)
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The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.
Planar Drawings and Angular Resolution: Algorithms and Bounds (Extended Abstract)
 IN PROC. 2ND ANNU. EUROPEAN SYMPOS. ALGORITHMS
, 1994
"... We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on th ..."
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Cited by 24 (5 self)
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We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on the angular resolution of planar straightline drawings, and show a continuous tradeoff between the area and the angular resolution. We also give lineartime algorithms for constructing planar straightline drawings with high angular resolution for various classes of graphs, such as seriesparallel graphs, outerplanar graphs, and triangulations generated by nested triangles. Our results are obtained by new techniques that make extensive use of geometric constructions.
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and pro ..."
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Cited by 19 (3 self)
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Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)area planar upward grid drawing that preserves the lefttoright ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...
Proximity Constraints and Representable Trees
, 1995
"... This paper examines an infinite family of proximity drawings of graphs called open and closed fidrawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the wellknown Gabriel, relative neighborhood and ..."
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Cited by 19 (10 self)
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This paper examines an infinite family of proximity drawings of graphs called open and closed fidrawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the wellknown Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit open fidrawings for 0 fi ! fi ! 1 or closed fidrawings for 0 fi ! fi 1 are given, as well as partial characterizations for other values of fi. For the intervals of fi in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed fidrawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.
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
A Framework for Drawing Planar Graphs with Curves and Polylines
 J. Algorithms
, 1998
"... We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well ..."
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Cited by 15 (3 self)
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We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any nvertex planar graph in an O(n) O(n) grid using polylines that have at most two bends per edge and asymptoticallyoptimal worstcase angular resolution. More significantly, we show how to adapt this algorithm to draw any nvertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. Al...
The Strength of Weak Proximity
, 1996
"... This paper initiates the study of weak proximitydrawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximitydrawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other poi ..."
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Cited by 13 (6 self)
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This paper initiates the study of weak proximitydrawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximitydrawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment(p# q) is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximitydrawing in which such segments must appear in the drawing. Most previously studied proximity regions are associated with a parameter fi,0 fi 1.For fixed fi,weak fidrawability is at least as expressive as strong fidrawability,as a strong fidrawing is also a weak one. Wegive examples of graph families and fi values where the two notions coincide, and a situation in which it is NPhard to determine weak fidrawability. On the other hand, wegive situations where weak proximity significantly increases the expressivepower of fidrawability: weshowthatevery graph has, for all sufficiently small fi,aweak fiproximitydrawing that is computable in linear time, and we show that every tree has, for every fi less than 2, a weak fidrawing that is computable in linear time.