Minimum-width grid drawings of plane graphs

Abstract

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) × ( n − 2) grid (for n ≥ 3), and that no grid smaller than ( 2n 3 − 1) × ( 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) 3 ⌟ , 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 ⌞ 2(n − 1) 3 ⌟ . The height of the produced drawings is bounded by 4⌞ 2(n − 1) 3 ⌟ − 1 . Our algorithm runs in linear time and is easy to implement.