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 the 3SAT reduction framework which 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-hard. 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-hard if a maximum of two bends per edge is allowed. We comment 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.