Two Algorithms for Three Dimensional Orthogonal Graph Drawing

We use basic results from graph theory to design two algorithms for constructing 3-dimensional, intersection-free orthogonal grid drawings of n vertex graphs of maximum degree 6. Our first algorithm gives drawings bounded by an O(x/~ ) • O(x/~ • O(v~ ) box; each edge route contains at most 7 bends. The best previous result generated edge routes containing up to 16 bends per route. Our second algorithm gives drawings having at most 3 bends per edge route. The drawings he in an O(n) • O(n) x O(n) bounding box. Together, the two algorithms initiate the study of bend/bounding box trade-off issues for 3-dimensional grid drawings. 1 I n t r o d u c t i o n The 3-dimensional orthogonal grid consists of grid points whose coordinates are all integers, together with the axis-parallel grid lines determined by these points. A 3-dimensional orthogonal grid drawing of a graph G places the vertices of G at grid points and routes the edges of G along sequences of contiguous segments contained in the grid lines. Edge routes are allowed to contain bends but are not allowed to cross or to overlap, i.e., no internal point, not necessarily a grid point, of one edge route may lie in any other edge route. Throughout this paper, grid refers to the 3-dimensional orthogonal grid, and grid drawing refers to the type of 3-dimensional orthogonal grid drawing just described. Note that because each grid point lies at the intersection of three grid lines, any graph that admits a grid drawing necessarily has maximum vertex degree at most 6. Figure 1 shows a grid drawing of a graph. This particular drawing lies in a 2 x 3 x 2 hounding box. The edge route joining the two extremal vertices in the Z-direction lies along the top, back and bottom faces of the box and contains 2 bends. The edge route joining the two extremal vertices in the X-direction also contains 2 bends, but passes through the interior of the box. While the graph drawing literature has extensively investigated 2-dimensional grid drawings of graphs (see [7]), 3-dimensional grid drawing has been little studied. Our research is motivated in part by recent interest in exploring the utility of 3-dimensional drawings of graphs for visualization purposes. It should also be