Optimal three-dimensional orthogonal graph drawing in the general position model

  • Authors:

  • David R. Wood

  • Affiliations:

  • School of Computer Science, Carleton University, Ottawa, Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

Let G be a graph with maximum degree at most six. A three-dimensional orthogonal drawing of G positions the vertices at grid-points in the three-dimensional orthogonal grid, and routes edges along grid lines such that edge routes only intersect at common end-vertices. In this paper, we consider three-dimensional orthogonal drawings in the general position model; here no two vertices are in a common grid-plane. Minimising the number of bends in an orthogonal drawing is an important aesthetic criterion, and is NP-hard for general position drawings. We present an algorithm for producing general position drawings with an average of at most 2 2/7 bends per edge. This result is the best known upper bound on the number of bends in three-dimensional orthogonal drawings, and is optimal for general position drawings of K7. The same algorithm produces drawings with two bends per edge for graphs with maximum degree at most five; this is the only known non-trivial class of graphs admitting two-bend drawings.