Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions

This paper investigates the following question: Given a grid φ, where φ is a proper subset of the integer 2D or 3D grid, which graphs admit straight-line crossing-free drawings with vertices located at (integral) grid points of φ? We characterize the trees that can be drawn on a strip, i.e., on a two-dimensional n × 2 grid. For arbitrary graphs we prove lower bounds for the height k of an n × k grid required for a drawing of the graph. Motivated by the results on the plane we investigate restrictions of the integer grid in 3D 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. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to an n× 2× 2 integer grid, called a box, does not admit the entire class of planar graphs. Communicated by: P. Mutzel and M. Jünger; submitted May 2002; revised October 2003. Research supported in part by: the MIUR project “Progetto ALINWEB: Algoritmica per Internet e per il Web”; and by the Natural Sciences and Engineering Research Council of Canada. Felsner et al., Restricted Integer Grids, JGAA, 7(4) 363–398 (2003) 364