Three dimensional graph drawing with fixed vertices and one bend per edge

A three-dimensional grid-drawing of a graph represents the vertices by distinct points in Z (called grid-points), and represents each edge as a polyline between its endpoints with bends (if any) also at gridpoints, such that distinct edges only intersect at common endpoints, and each edge only intersects a vertex that is an endpoint of that edge. This topic has been previously studied in [1–5]. We focus on the problem of producing such a drawing, where the vertices are fixed at given grid-points. This variant has been previously studied in [5, 6]. Meijer and Wismath [5] recently proved the following theorem: