Rectangle-of-Influence Drawings of Four-Connected Plane Graphs

A rectangle-of-in uence drawing of a plane graph G is a straight-line planar drawing of G such that there is no vertex in the proper inside of the axis-parallel rectangle de ned by the two ends of any edge. In this paper, we show that any 4-connected plane graph G with four or more vertices on the outer face has a rectangle-of-in uence drawing in an integer grid such that W + H n, where n is the number of vertices in G, W is the width and H is the height of the grid. Thus the areaW H of the grid is at most d(n 1)=2e b(n 1)=2c. Our bounds on the grid sizes are optimal in a sense that there exist an in nite number of 4connected plane graphs whose drawings need grids such that W +H = n 1 and W H = d(n 1)=2e b(n 1)=2c. We also show that the drawing can be found in linear time.