Convex grid drawings of four-connected plane graphs

A convex grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on grid points, all edges are drawn as straight-line segments between their endpoints without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of any 4-connected plane graph G with four or more vertices on the outer face boundary. The algorithm yields a drawing in an integer grid such that W + H less than or equal to n - 1 if G has n vertices, where W is the width and H is the height of the grid. Thus the area W x H of the grid is at [(n - 1)/2].[(n - 1)/2]. Our bounds on the grid sizes are optimal in the sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - I and W x H = [(n - 1)/2] . [(n - 1)/2].