A Graph-Theretic Framework for Unique Foldability in Rigid Origami

Rigid origami studies foldable structures composed of rigid faces con- nected by hinges, where foldability is governed by geometric constraints on dihedral angles. While many classical models focus on counting degrees of freedom or ensuring consistency of local folding conditions, less atten- tion has been given to systems in which a single hinge angle uniquely determines the entire global configuration. In this paper, we introduce and formalize Koyejun circuits, a class of rigid origami hinge networks exhibiting a strong global determinacy property: fixing the dihedral an- gle of any one hinge uniquely determines the dihedral angles of all oth- ers. We develop a propagation-based framework that models how local vertex constraints transmit angular information across the hinge graph, and we derive a structural characterization of one-degree-of-freedom rigid origami systems. We then analyze open triangulated Koyejun circuits, de- fine a hierarchy of indirect propagation orders, and prove that admissible expansions preserving global determinacy are uniquely constrained to a specific fan-type vertex extension. This yields a combinatorial invariant relating the numbers of vertices, edges, and faces, establishing a neces- sary graph-theoretic condition for such circuits. Finally, we investigate closed Koyejun circuits, corresponding to hypothetical polyhedral rigid origami structures with global one-hinge determinacy. We prove a non- existence theorem showing that no closed polyhedral structure can satisfy the Koyejun circuit property, due to unavoidable overconstraint arising from boundary closure. Our results provide a new graph-theoretic and geometric perspective on rigidity, determinacy, and information propaga- tion in rigid origami, offering a foundation for future studies of globally constrained foldable mechanisms.