Discrete 4D printing

4D printing allows one to fabrication 3D objects from patterned, 2D sheets by locally growing a material. Finding the optimal shape from a pattern of growth is challenging, owing to the fact that there may be many solutions (or none) that can accommodate prescribed growth. Here, we consider a discrete analogue of this problem: how many 3D configurations can accommodate the prescribed growth of the length of edges of a graph and how does that depend on the topology of the graph? In our analysis, propose an optimal discrete structure for which it is computationally easier to find shape given edge lengths and propose a way to count possible solutions.