Dual graphs with 3 vertices (in red or black, along with one example of a corresponding existing RNA structure) used to generate dual graphs with 4 vertices by connecting one additional vertex (in blue). A total of 31 edge combinations (generated as described in Subsection 2.2.2) used to connect the new vertex to any of the 3 previous vertices with 1, 2, 3, or 4 edges are also listed. The numbers under columns V1, V2, and V3 in each edge combination are equal to the number of edges connecting the new and previous vertex numbers 1, 2, and 3 respectively. These 31 edge combinations are used to generate 31 4-vertex graphs, 25 of which (edge combinations highlighted in blue) follow all dual graph rules (shown in Figure 4). The remaining 6 edge combinations (in black) generate graphs that are discarded (shown in Supplementary Figure S1).