Dodecahedron
This structure has 20 vertices and 30 edges. Because each vertex has a degree of three, this structure cannot be threaded without adding augmenting edges to the structure so that every vertex has an even degree. The figure below on the left shows the augmenting edges (light blue) added to the graph of original edges (dark blue). The figure to its right illustrates the subsequent 5-way symmetrical circuit that can be traversed.
In the following diagram, the black arrows represent the Euler circuit that the scaffolding strand would follow through the structure. Though the staple strand arrows are left out of the image for clarity, the staples would clearly follow in the opposite direction to the scaffolding strand, holding the DNA in place at the vertices.
Threading sequence: 1, 8, 7, 17, 16, 7, 6, 1, 5, 6, 15, 16, 20, 15, 14, 5, 4, 14, 13, 20, 19, 13, 4, 3, 12, 11, 19, 18, 11, 10, 3, 2, 10, 9, 18, 17, 9, 8, 2, 1
As a result of this 5-way symmetry of the Euler circuit and augmenting edges, there are only 4 unique vertex configurations necessary to self-assemble the dodecahedron. The directions of the staple strands (red) are shown along with the scaffolding strand (black) in the closer look at these different configurations:
