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 kno...

• Type: Journal
• Year: 2006
• Where: ORDER
• Authors: Vida Dujmovic, David R. Wood
• Keywords: ORDER 2006, Three-dimensional Grid, Three-dimensional Grid Drawings, Upward Three-dimensional Grid