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-o 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. ? 2000 Elsevier Science B.V. All rights reserved.