# A Comparison of Different Folding Models in Variations of the Map Folding Problem
## 6 Performances of Three Folding Models in Boundary VOP
### Strict Inclusion Relations among Three Folding Models  
As aforementioned, the strict inclusion relation among the folding models in Total VOP follows their strict inclusion relation in Boundary VOP. It is clear that every reachable flat-folded state of the simple folding model is definitely reachable for the simple folding-unfolding model, and every reachable state of the latter one is also reachable for the general folding model. Then, to prove the strict inclusion relation in Boundary VOP, we only have to prove that there do exist reachable boundary overlapping orders of the general folding model that cannot be reached by the simple folding-unfolding model, as well as reachable boundary overlapping order of the simple folding-unfolding model that cannot be accessed by a simple folding model. In the following, we provide concrete instances that respectively belong to the difference sets of their reachable boundary overlapping orders. For the difference between the general folding model and the simple folding-unfolding model, an instance is given in Figure 7a. The total overlapping order on the left side was first proposed in [7] as an instance that needs complex general folds (not simple folds). Its corresponding boundary overlapping order is given in Figure 7a on the right side. In this instance, since only the center square labeled five in the left figure is not a boundary square, the boundary overlapping order decides the entire MV assignment. Clearly, there exists no crease line entirely assigned to valleys or mountains in this figure, which means that a first simple fold could not be applied. Thus, this instance corresponds to a reachable boundary overlapping order of the general folding model that cannot be reached by the simple folding-unfolding model. However, even in an MV-assignment, including crease lines entirely assigned the same, there still exist boundary overlapping orders reachable by the general folding model but unreachable by the simple folding-unfolding model. Such a boundary overlapping order of the smallest map (of size 2 \(\times\) 3) is given on the left side of Figure 7b, where the boundary overlapping order is also the total overlapping order. For the maps with larger \(m\) and \(n\), the result still holds. An instance is given on the right side of Figure 7b. The above instances naturally bring this question: what kind of boundary overlapping order reachable by the general folding model can also be reached by the simple folding-unfolding model? We give the answer in Theorem 4. **Theorem 4**. : _A valid boundary overlapping order of the general folding model is also valid for the simple folding-unfolding model if and only if every flat-folded state can be separated to rectangles with respect to the order._  
Proof of Theorem 4.: Considering a valid folding sequence by the simple folding-unfolding model from the last step, the reverse sequence can be described as each time applying all the unfolds and folds along the same direction until no more unfolds along this direction can be applied, then changing the direction and repeating the unfolds and folds. During the process, every fold or unfold is applied on a rectangle shape, and the rectangle would be divided into smaller rectangles by the crease lines. Thus, it is clear that every flat-folded state of the simple folding-unfolding model can be separated into rectangles. Conversely, as long as at each partly folded-state after the direction change can be separated to rectangles before the next time of direction change, the map can be viewed as a 1 \(\times\)\(n\) map where the overlapping of the rectangles can be viewed as the overlapping of line segments. Referencing the result that every partly folded state of a 1 \(\times\)\(n\) map is also reachable by a simple folding-unfolding model, the reverse sequence can be achieved. This ensures that if every flat-folded state can be separated into rectangles, the overlapping by the general folding model must also be valid for the simple folding-unfolding model.