Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings

Abstract

This paper presents the first non-trivial lower bounds for the total number of bends in 3-D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3-D 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 c-connected simple graphs, multigraphs, and pseudographs (2 ≤ c ≤ 6) of maximum degree ∆ (3 ≤ ∆ ≤ 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 3-D orthogonal graph drawings. These results have significant ramifications for the `2-bends problem', which is one of the most important open problems in the field.