Realization and Connectivity of the Graphs of Origami Flat Foldings

TL;DR

This paper characterizes the connectivity properties of graphs derived from flat origami foldings, establishing conditions under which such graphs can be realized and their connectivity constraints.

Contribution

It provides a complete characterization of the graphs of flat foldings, including conditions for trees and connectivity requirements for general folding patterns.

Findings

Trees can be realized as folding graphs if all internal vertices have even degree greater than two.

Folding graphs must be 2-vertex-connected and 4-edge-connected.

The results apply to unbounded sheets with a vertex at infinity.

Abstract

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.