Three-dimensional Grid Drawings Research Articles

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 $\mathcal{O}\left({n^{3/2}}\right)$ 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.

The Maximum Number of Edges in a Three-Dimensional Grid-Drawing

An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. The first universal lower bound on the volume of threedimensional grid-drawings is obtained as a corollary. Our results generalise to the setting of multi-dimensional polyline grid-drawings.

Three-Dimensional 1-Bend Graph Drawings

We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is ( cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n 3 = log 2 n) volume and one bend per edge. The best previous bound was O(n 3 ).

Three-dimensional orthogonal graph drawing algorithms

We use basic results from graph theory to design algorithms for constructing three-dimensional, intersection-free orthogonal grid drawings of n vertex graphs of maximum degree 6. The best previous result generated a drawing bounded by an O( n )× O( n )× O( n ) box, with each edge route containing up to 16 bends. Our algorithms initiate the study of bend/bounding box trade-off issues for three-dimensional grid drawings and produce drawings with the following characteristics: 1. at most 7 bends per edge route, bounded by an O( n )× O( n )× O( n ) box; 2. at most 6 bends per edge route, bounded by an O( n )× O( n )× O(n) box; 3. at most 5 bends per edge route, bounded by an O( n )× O(n)× O(n) box; 4. at most 3 bends per edge route, bounded by an O( n)×O( n)×O( n) box; and 5. for maximum degree 4 graphs, at most 3 bends per edge route, bounded by and O( n)×O( n)×O(1) box.