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25
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.
A Technique for Drawing Directed Graphs
 IEEE Transactions on Software Engineering
, 1993
"... We describe a fourpass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce cros ..."
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Cited by 222 (19 self)
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We describe a fourpass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce crossings. The third pass finds optimal coordinates for nodes by constructing and ranking an auxiliary graph. The fourth pass makes splines to draw edges. The algorithm makes good drawings and runs fast. 1.
Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
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Cited by 65 (0 self)
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We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
A Better Heuristic for Orthogonal Graph Drawings
 COMPUT. GEOM. THEORY APPL
, 1998
"... An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is ben ..."
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Cited by 61 (6 self)
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An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is bent at most twice. In particular for nonplanar and nonbiconnected planar graphs, this is a big improvement. The algorithm is very simple, easy to implement, and it handles both planar and nonplanar graphs at the same time.
MinimumWidth Grid Drawings of Plane Graphs
 Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science
, 1995
"... Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straightline segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each pl ..."
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Cited by 31 (11 self)
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Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straightline segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each plane graph can be drawn in such a way in a (n \Gamma 2) \Theta (n \Gamma 2) grid (for n 3), and that no grid smaller than (2n=3 \Gamma 1) \Theta (2n=3 \Gamma 1) can be used for this purpose, if n is a multiple of 3. In fact, for all n 3, each dimension of the resulting grid needs to be at least b2(n \Gamma 1)=3c, even if the other one is allowed to be unbounded. In this paper we show that this bound is tight by presenting a grid drawing algorithm that produces drawings of width b2(n \Gamma 1)=3c. The height of the produced drawings is bounded by 4b2(n \Gamma 1)=3c \Gamma 1. Our algorithm runs in linear time and is easy to implement. 1 Introduction The problem of automatic graph drawing ha...
Embedding vertices at points: Few bends suffice for planar graphs
 in Graph Drawing (Proc. GD '99), LNCS 1731
, 2002
"... The existing literature gives ecient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general ..."
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Cited by 28 (1 self)
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The existing literature gives ecient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping fourconnected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NPcomplete to decide whether there is an mapping such that each edge has at most one bend.
Fractal Approaches for Visualizing Huge Hierarchies
 In Proceedings of the 1993 IEEE Symposium on Visual Languages
, 1993
"... This paper describes fractal approaches to the problems which associate with visualizing huge hierarchies. The geometrical characteristic of a fractal, selfsimilarity, allows users to visually interact with a huge tree in the same manner at every level of the tree. The fractal dimension, a measure o ..."
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Cited by 23 (1 self)
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This paper describes fractal approaches to the problems which associate with visualizing huge hierarchies. The geometrical characteristic of a fractal, selfsimilarity, allows users to visually interact with a huge tree in the same manner at every level of the tree. The fractal dimension, a measure of complexity, makes it possible to control the total amount of displayed nodes. A prototype visualization system for UNIX directories is also shown. 1 Introduction Visualization systems for hierarchical structures, especially for huge 1 data structures, have a potential usefulness. For example, the visualization of whole UNIX directories might help system administrators to maintain the file systems. Since administrators could recognize, through the visualization, local file systems of each computer and remote file systems mounted by using NFS (Network File System), they might avoid mistakes, such as deleting or moving files which are being referenced by other computers. It is, however, m...
Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 23 (11 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices.
Two Algorithms for Finding Rectangular Duals of Planar Graphs
, 1992
"... We present two lineartime algorithms for computing a regular edge labeling of 4connected planar triangular graphs. This labeling is used to compute in linear time a rectangular dual of this class of planar graphs. The two algorithms are based on totally different frameworks, and both are concep ..."
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Cited by 11 (3 self)
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We present two lineartime algorithms for computing a regular edge labeling of 4connected planar triangular graphs. This labeling is used to compute in linear time a rectangular dual of this class of planar graphs. The two algorithms are based on totally different frameworks, and both are conceptually simpler than the previous known algorithm and are of independent interests. The first algorithm is based on edge contraction. The second algorithm is based on the canonical ordering. This ordering can also be used to compute more compact visibility representations for this class of planar graphs.
Lower Bounds for Planar Orthogonal Drawings of Graphs
 Inform. Process. Lett
, 1994
"... We study planar orthogonal drawings of graphs and provide lower bounds on the number of bends along the edges. We exhibit graphs on n vertices that require \Omega\Gamma n) bends in any layout, and show that there exist optimal drawings that require\Omega\Gamma n) bends and have all of them on a si ..."
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Cited by 8 (2 self)
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We study planar orthogonal drawings of graphs and provide lower bounds on the number of bends along the edges. We exhibit graphs on n vertices that require \Omega\Gamma n) bends in any layout, and show that there exist optimal drawings that require\Omega\Gamma n) bends and have all of them on a single edge of length\Omega\Gamma n 2 ). This work finds applications in VLSI layout, aesthetic graph drawing, and communication by light or microwave. Keywords: planar embedding, VLSI layout, orthogonal drawing, bends. November 26, 1994 Research supported in part by the National Science Foundation under grant CCR9007851, by the U.S. Army Research Office under grant DAAL0391G0035, by the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N0001483K0146 and ARPA order 6320, amendment 1, and by Cadre Technologies, Inc. y Research supported in part by the Texas Advanced Research Program under Grant No. 3972. z Research supported in part by th...