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We investigate the graphs formed from the vertices and creases of an origamipattern that can be folded flat along all of its creases. As we show, this ispossible for a tree if and only if the internal vertices of the tree all haveeven degree greater than two. However, we prove that (for unbounded sheets ofpaper, with a vertex at infinity representing a shared endpoint of all creasedrays) the graph of a folding pattern must be 2-vertex-connected and4-edge-connected.

Realization and Connectivity of the Graphs of Origami Flat Foldings