Convex Grid Drawings of Four-Connected Plane Graphs

Abstract

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 ≤ n - 1 if G has n vertices, where W is the width and H is the height of the grid. Thus the area W × H of the grid is at most ⌈(n - 1)/2⌉ ċ ⌊(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 - 1 and W × H = ⌈(n - 1)/2⌉ ċ ⌊(n - 1)/2⌋.