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34
An Experimental Comparison of Four Graph Drawing Algorithms
, 1995
"... In this paper we present an extensive experimental study comparing four generalpurpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in so ..."
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Cited by 62 (9 self)
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In this paper we present an extensive experimental study comparing four generalpurpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in "reallife" software engineering and database applications. The experiments
Convex Grid Drawings of 3Connected Planar Graphs
, 1994
"... We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane gr ..."
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Cited by 37 (7 self)
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We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane graph G (with n 3), finds such a straightline convex embedding of G into a (n \Gamma 2) \Theta (n \Gamma 2) grid. 1 Introduction In this paper we consider the problem of aesthetic drawing of plane graphs, that is, planar graphs that are already embedded in the plane. What is exactly an aesthetic drawing is not precisely defined and, depending on the application, different criteria have been used. In this paper we concentrate on the two following criteria: (a) edges should be represented by straightline segments, and (b) faces should be drawn as convex polygons. F'ary [6], Stein [14] and Wagner [18] showed, independently, that each planar graph can be drawn in the plane in such a way that ...
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...
Convex Drawings of Graphs in Two and Three Dimensions
, 1996
"... We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) × (3n/2) area under the edge L 1 resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution ru ..."
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Cited by 29 (10 self)
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We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) × (3n/2) area under the edge L 1 resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) × O(n) area under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers; ffl 3D convex drawings with O(1) × O(1) × O(n) volume under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers. We also
Upward Planar Drawing of Single Source Acyclic Digraphs
, 1990
"... A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We ..."
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Cited by 15 (1 self)
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A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given singlesource acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...
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...
Planar Drawings of Plane Graphs
, 2000
"... this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results. ..."
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Cited by 13 (3 self)
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this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.
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.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Optimal Polygonal Representation of Planar Graphs
"... Abstract. In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched ..."
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Cited by 10 (9 self)
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Abstract. In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges with slopes 0, 1,1. 1