Results 1 
3 of
3
Lower Bounds for the Number of Bends in ThreeDimensional Orthogonal Graph Drawings
, 2003
"... This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tigh ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of cconnected simple graphs, multigraphs, and pseudographs (2 ≤ c ≤ 6) of maximum degree Δ (3 ≤ Δ ≤ 6), with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3D orthogonal graph drawings. These results have significant ramifications for the `2bends problem', which is one of the most important open problems in the field.
Minimising the Number of Bends and Volume in 3Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout
, 2004
"... A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3d ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
A 3dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at gridpoints in the 3dimensional orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. Minimising the number of bends and the volume of 3dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, socalled diagonal drawings. This vertexlayout strategy was introduced in the 3BENDS algorithm of Eades et al. [Discrete Applied Math. 103:5587, 2000]. We show that minimising the number of bends in a diagonal drawing of a given graph is NPhard. 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 vertexordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the abovementioned 3BENDS algorithm, produces 3bend drawings with o(n ) volume, which is the best known upper bound for the volume of 3dimensional orthogonal graph drawings with at most three bends per edge.