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 $\text{(}\text{2n}\text{3}\text{− 1) × (}\text{2n}\text{3}\text{− 1)}$ 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 $\text{⌞}\text{2(n − 1)}\text{3}\text{⌟}$, 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 $\text{⌞}\text{2(n − 1)}\text{3}\text{⌟}$. The height of the produced drawings is bounded by $\text{4⌞}\text{2(n − 1)}\text{3}\text{⌟ − 1}$. Our algorithm runs in linear time and is easy to implement.