Minimum-Width Grid Drawings of Plane Graphs

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 an (n 2) x (n 2) grid (for n ~> 3), and that no grid smaller than (2n/3 1) x (2n/3 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 [2(n 1)/3J, even if the other one is allowed to be unbounded. In this paper we show that this bound is fight by presenting a grid drawing algorithm that produces drawings of width 12(n 1)/33. The height of the produced drawings is bounded by 4/2(n 1)/3J 1. Our algorithm runs in linear time and is easy to implement. ¢ 1998 Elsevier Science B.V. All rights reserved.