Straight - Line Drawings on Restricted

This paper investigates the following question: Given an integer grid , where is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossing-free drawings with vertices located at the grid points of ? We characterize the trees that can be drawn on a two dimensional c n k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of innnite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal { it supports all outerplanar graphs of n vertices. This is the rst algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n 2 2 integer grid, called a box, does not admit the entire class of planar graphs.