<p>One recent application of graph theory is the theoretical analysis of self-assembling DNA nanostructures. In a laboratory setting, branched junction molecules of DNA can self-assemble into a variety of geometric structures, a technology pioneered by Seeman’s laboratory in the 1980s. Theoretical efficiency of this process can be increased by creative, mathematical design. One aim is to minimize the number of distinct molecule types required to achieve a target structure via self-assembly. A second, more difficult, aim is to avoid the formation of unwanted substructures. Following the flexible tile-based model introduced by Ellis-Monaghan et al. (In: Discrete and Topological Models in Molecular Biology, Springer, Cham, pp. 241–270, 2014), branched junction molecules can be formally represented as vertices with extending half-edges and target structures as discrete graphs. Here we present, within the context of this model, provably optimal solutions in three different levels of laboratory restriction for assembly targets resembling dipyramids and trapezohedrons. Some of our results highlight straightforward application of results from the existing literature, while others illustrate the mathematical challenges that can arise.</p>