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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 ..."
Abstract

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...
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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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.