Minimising the Number of Bends and Volume in 3-Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout

• Authors: David R. Wood
• Venue: Algorithmica
• Year: 2004

A 3-dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at grid-points in the 3-dimensional orthogonal grid, and routes edges along grid-lines such that edge routes only intersect at common end-vertices. Minimising the number of bends and the volume of 3-dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3-dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, so-called diagonal drawings. This vertex-layout strategy was introduced in the 3-BENDS algorithm of Eades et al. [Discrete Applied Math. 103:55–87, 2000]. We show that minimising the number of bends in a diagonal drawing of a given graph is NP-hard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal in linear time. Using two heuristics for determining this vertex-ordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the above-mentioned 3-BENDS algorithm, produces 3-bend drawings with n3+o(n3) volume, which is the best known upper bound for the volume of 3-dimensional orthogonal graph drawings with at most three bends per edge.