Three-Dimensional Orthogonal Drawing Research Articles

Complexity results for three-dimensional orthogonal graph drawing

In this paper we consider the problem of finding three-dimensional orthogonal drawings of maximum degree six graphs from the computational complexity perspective. We introduce a 3SAT reduction framework that can be used to prove the NP-hardness of finding three-dimensional orthogonal drawings with specific constraints. By using the framework we show that, given a three-dimensional orthogonal shape of a graph (a description of the sequence of axis-parallel segments of each edge), finding the coordinates for nodes and bends such that the drawing has no intersection is NP-complete. Conversely, we show that if node coordinates are fixed, finding a shape for the edges that is compatible with a non-intersecting drawing is a feasible problem, which becomes NP-complete if a maximum of two bends per edge is allowed. We comment on the impact of these results on the two open problems of determining whether a graph always admits a drawing with at most two bends per edge and of characterizing orthogonal shapes admitting an orthogonal drawing without intersections.

Three-Dimensional Orthogonal Graph Drawing with Optimal Volume

An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axis-aligned boxes, and edges represented by paths in the grid that only possibly intersect at common endpoints. In this paper we study three-dimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where the vertices have a bounded aspect ratio, (2) drawings where the surfaces of vertices are proportional to their degrees, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that in all scenarios match the lower bounds within a constant factor.

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

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 227 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.