Grid Drawings of Four-Connected Plane Graphs

A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face. The algorithm takes time O(n) and needs a rectangular grid of width \lceil n/2 \rceil - 1 and height \lceil n/2 \rceil if G has n vertices. The algorithm is best possible in the sense that there are an infinite number of 4-connected plane graphs any grid drawings of which need rectangular grids of width \lceil n/2 \rceil - 1 and height \lceil n/2 \rceil