Finding Some Impossibility of Flat-Folding of Given Origami Crease Pattern by Graphical Representation

Abstract

The flat-foldability problem in origami asks whether a given crease pattern can be folded flat without any physical penetration or intrusion of polygons into the creases. As established by Bern and Hayes, determining the global flat-foldability of a general crease pattern is an NP-hard problem. In this paper, we focus on unsigned crease patterns that satisfy the necessary local conditions imposed by the Kawasaki-Justin theorem at all interior vertices. To evaluate global foldability, we introduce an undirected graph representation-an overlap graph-that models pairwise non-intrusion constraints among overlapping polygons in a flattened state. Using this graphical representation, we propose a polynomial-time algorithm to efficiently detect the impossibility of flat-folding by analyzing the algebraic properties of the graph's cycle basis. Specifically, we classify the nodes (intermediations) along each cycle and prove that the parity of a specific node kind governs the mathematical consistency of the loop. Detecting a self-inconsistent, frustrated, cycle via parity evaluation provides a robust sufficient condition for demonstrating that the entire crease pattern cannot be flat-folded. This result successfully isolates the tractable components of flat-foldability from its worst-case NP-hardness, providing a deeper understanding of the precise structural features that cause global computational difficulty. We also demonstrate the efficacy of our method by applying it to a well-known crease pattern that is fundamentally impossible to flat-fold.