An example of geometric origami design with benefit of graph enumeration algorithms

This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from brute force enumeration, up to the equivalence to a dedicated problem in graph theory. This leads to algorithms starting from an untractable case on modern computers, up to a run of few seconds on a portable personal computer. This emphasizes the need for a prior analysis by humans before considering the assistance of computers for complex design problems. The graph problem is an enumeration of spanning trees from a grid graph, leading to a coarse scale description of the topology of the paper edge on the flat-folded state.