Upward Three-Dimensional Grid Drawings of Graphs

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every n-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O (n3/2) volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the z-direction. We prove that every dag has an upward three-dimensional grid drawing with O (n3) volume, which is tight for the complete dag. The previous best upper bound was O (n4). Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O(n²) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n³) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges