Upward Three-Dimensional Grid Drawings of Graphs

Abstract

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 $\mathcal{O}\left({n^{3/2}}\right)$ 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 $\textsf{z}$ -direction. We prove that every dag has an upward three-dimensional grid drawing with $\mathcal{O}\left({n^3}\right)$ volume, which is tight for the complete dag. The previous best upper bound was $\mathcal{O}\left({n^4}\right)$ . Our main result concerning upward drawings is that every $c$ -colourable dag ( $c$ constant) has an upward three-dimensional grid drawing with $\mathcal{O}\left({n^2}\right)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $\mathcal{O}\left({n^3}\right)$ 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.