  Three-dimensional shells can be synthesized from the spontaneous self-folding
of two-dimensional templates of interconnected panels, called nets. However,
some nets are more likely to self-fold into the desired shell under random
movements. The optimal nets are the ones that maximize the number of vertex
connections, i.e., vertices that have only two of its faces cut away from each
other in the net. Previous methods for finding such nets are based on random
search and thus do not guarantee the optimal solution. Here, we propose a
deterministic procedure. We map the connectivity of the shell into a shell
graph, where the nodes and links of the graph represent the vertices and edges
of the shell, respectively. Identifying the nets that maximize the number of
vertex connections corresponds to finding the set of maximum leaf spanning
trees of the shell graph. This method allows not only to design the
self-assembly of much larger shell structures but also to apply additional
design criteria, as a complete catalog of the maximum leaf spanning trees is
obtained.
