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A Lineartime Algorithm for Drawing a Planar Graph on a Grid
 Information Processing Letters
, 1989
"... We present a lineartime algorithm that, given an nvertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straightline segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid i ..."
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Cited by 37 (5 self)
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We present a lineartime algorithm that, given an nvertex planar graph G, finds an embedding of G into a (2n \Gamma 4) \Theta (n \Gamma 2) grid such that the edges of G are straightline segments. 1 Introduction We consider the problem of embedding the vertices of a planar graph into a small grid in the plane in such a way that the edges are straight, nonintersecting line segments. The existence of such straightline embeddings for planar graphs was independently discovered by F'ary [Fa48], Stein [St51], and Wagner [Wa36]; this result also follows from Steinitz's theorem on convex polytopes in three dimensions [SR34]. The first algorithms for constructing straightline embeddings [Tu63, CYN84, CON85] required highprecision arithmetic, and the resulting drawings were not very aesthetic, since they tend to produce uneven distributions of vertices over the drawing area. Rosenstiehl and Tarjan [RT86] noticed that it would be convenient to be able to map veritices of a planar graph into a...
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 30 (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...
Comparing and Evaluating Layout Algorithms within GraphEd
 J. Visual Languages and Computing
, 1995
"... This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd ..."
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Cited by 16 (2 self)
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This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd
An Experimental Comparison of Three Graph Drawing Algorithms (Extended Abstract)
, 1995
"... In this paper we present an extensive experimental study... ..."
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Cited by 16 (5 self)
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In this paper we present an extensive experimental study...
Drawing Directed Acyclic Graphs: An Experimental Study
, 1996
"... In this paper we consider the class of directed acyclic graphs (DAGs), and present the results of an experimental study on four drawing algorithms specifically developed for DAGs. Our study is conducted on two large test suites of DAGs and yields more than 30 charts comparing the performance of ..."
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Cited by 9 (0 self)
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In this paper we consider the class of directed acyclic graphs (DAGs), and present the results of an experimental study on four drawing algorithms specifically developed for DAGs. Our study is conducted on two large test suites of DAGs and yields more than 30 charts comparing the performance of the drawing algorithms with respect to several quality measures, including area, crossings, bends, and aspect ratio. The algorithms exhibit various tradeoffs with respect to the quality measures, and none of them clearly outperforms the others.
Advances in the Theory and Practice of Graph Drawing
 Theor. Comp. Sci
, 1996
"... The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph d ..."
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Cited by 4 (0 self)
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The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph drawing. Specific topics include bounds and tradeoffs for drawing properties, threedimensional representations, methods for constraint satisfaction, and experimental studies. 1 Introduction In this paper, we survey selected research trends in graph drawing, and overview some recent results of the author and his collaborators. Graph drawing addresses the problem of constructing geometric representations of graphs, a key component of support tools for complex applications in science and engineering. Graph drawing is a young research field that has growth very rapidly in the last decade. One of its distinctive characteristics is to have furthered collaborative efforts between computer scien...
A Note on Grid Drawings of Plane Graphs
, 1998
"... We consider the problem of drawing plane graphs such that the vertices are represented by grid points and edges are drawn as straight lines. It is an open problem to determine the minimum size ¯ \Theta of grids that admit straightline drawings of all nvertex plane graphs. Algorithms are known for ..."
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We consider the problem of drawing plane graphs such that the vertices are represented by grid points and edges are drawn as straight lines. It is an open problem to determine the minimum size ¯ \Theta of grids that admit straightline drawings of all nvertex plane graphs. Algorithms are known for producing drawings in grids of size (n \Gamma 2) \Theta (n \Gamma 2) and b2(n \Gamma 1)=3c \Theta b8(n \Gamma 1)=3c. The currently known lower bound is that each dimension satisfies ¯; b2(n \Gamma 1)=3c. In this note, we slightly improve this lower bound by showing that ¯ = b2(n \Gamma 1)=3c implies d2n=3e. Thus it is impossible to achieve the minimum width and height simultaneously.