  We investigate the graphs formed from the vertices and creases of an origami
pattern that can be folded flat along all of its creases. As we show, this is
possible for a tree if and only if the internal vertices of the tree all have
even degree greater than two. However, we prove that (for unbounded sheets of
paper, with a vertex at infinity representing a shared endpoint of all creased
rays) the graph of a folding pattern must be 2-vertex-connected and
4-edge-connected.
