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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 46 (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...
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 43 (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 32 (12 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...
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.
RectangleofInfluence Drawings of FourConnected Plane Graphs (Extended Abstract)
, 2005
"... A rectangleofinfluence drawing of a plane graph G is no vertex in the proper inside of the axisparallel rectangle defined by the two ends of any edge. In this paper, weshow that any 4connected plane graph G rectangleofinfluence drawing in an integer grid such that W + H n, where n is the numb ..."
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Cited by 2 (0 self)
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A rectangleofinfluence drawing of a plane graph G is no vertex in the proper inside of the axisparallel rectangle defined by the two ends of any edge. In this paper, weshow that any 4connected plane graph G rectangleofinfluence drawing in an integer grid such that W + H n, where n is the numberofvertices in G, W is the width and H is the height of the grid. Thus the area W \ThetaH of the grid is at most d(n;1)=2e\Delta b(n;1)=2c. Our bounds on the grid sizes are optimal in a sense that there exist an infinite number of 4connected plane graphs whose drawings need grids such that W +H = n;1andW \Theta H = d(n ; 1)=2e\Delta b(n ; 1)=2c. We also showthatthe drawing can be found in linear time.
Grid Embedding of Internally Triangulated Plane Graphs without Nonempty Triangles
, 1995
"... A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper, we show that, if an internally triangulated plane graph G has no nonempty triangles (a nonempty tri ..."
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Cited by 1 (0 self)
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A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper, we show that, if an internally triangulated plane graph G has no nonempty triangles (a nonempty triangle is a triangle of G containing some vertices in its interior), then G can be embedded on a grid of size W \Theta H such that W + H n, W (n + 3)=2 and H 2(n \Gamma 1)=3, where n is the number of vertices of G. Such an embedding can be computed in linear time. 1 Introduction Let G = (V; E) be a graph with n vertices. We always assume n 3 in this paper. G is planar if it can be drawn on the plane such that the vertices are located at distinct points, and the edges are represented by nonintersecting curves joining their endpoints. A plane graph is a planar graph with a fixed plane embedding. A straight line grid embedding of a planar graph is a drawing where the vertices are located at...
Convex Grid Drawings of Plane Graphs with . . .
, 2008
"... In a convex drawing of a plane graph, all edges are drawn as straightline segments without any edgeintersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally tric ..."
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In a convex drawing of a plane graph, all edges are drawn as straightline segments without any edgeintersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n â1) Ã(n â1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n Ã n 2 grid if T(G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. Our convex grid drawing has a rectangular contour, while most of the known algorithms produce grid drawings having triangular contours.
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.
On Drawing a Graph Convexly in the Plane (Extended Abstract)
"... ) ? Hristo N. Djidjev Department of Computer Science, Rice University, Hoston, TX 77251, USA Abstract. Let G be a planar graph and H be a subgraph of G. Given any convex drawing of H, we investigate the problem of how to extend the drawing of H to a convex drawing of G. We obtain a necessary and s ..."
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) ? Hristo N. Djidjev Department of Computer Science, Rice University, Hoston, TX 77251, USA Abstract. Let G be a planar graph and H be a subgraph of G. Given any convex drawing of H, we investigate the problem of how to extend the drawing of H to a convex drawing of G. We obtain a necessary and sufficient condition for the existence and a linear algorithm for the construction of such an extension. Our results and their corollaries generalize previous theoretical and algorithmic results of Tutte, Thomassen, Chiba, Yamanouchi, and Nishizeki. 1 Introduction The problem of embedding of a graph in the plane so that the resulting drawing has nice geometric properties has received recently significant attention. This is due to the large number of applications including circuit and VLSI design, algorithm animation, information systems design and analysis. The reader is referred to [1] for annotated bibliography on graph drawings. The first lineartime algorithm for testing a graph for plan...