TITLE: Complete folds and one cut
QUESTION [5 upvotes]: The fold-and-cut theorem states that any shape with straight sides can be cut by a single complete straight cut if the paper is the folded flat in the right way. Here is an example from an answer on another MO question:
     
Note that the folds are not complete, i.e. they are not along lines that run through the whole paper but stop somewhere in the middle. This makes it a little cumbersome to fold the pattern.
My questions are:

Is there any shape with straight edges that can't be obtained by complete folds and one complete cut? If yes, which kind of shapes can be obtained by complete folds and one complete cut?

REPLY [4 votes]: Not an answer. The obvious place to start is to fold-and-$1$-cut an irregular triangle,
whose natural method of cuts are incomplete in @Dirk's notation,
not simple folds in other notation1:

     


       

Figure from How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra.

       

Copied from MESE post Examples of Mathematical Beauty in School Mathematics
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1
Arkin, Esther M., Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph SB Mitchell, Saurabh Sethia, and Steven S. Skiena. "When can you fold a map?." Computational Geometry 29, no. 1 (2004): 23-46.
Journal link.