We present a linear-time algorithm that, given an n-vertex 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 straight-line 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, non-intersecting line segments. The existence of such straight-line 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 high-precision 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...

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 straight-line 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...

this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.

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 non-intersecting straight line segments. In this paper, we show that, if an internally triangulated plane graph G has no non-empty triangles (a non-empty 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...