Kirigami is a transformable system obtained by adding cuts on a sheet material and bending or folding the sheet. It is reported that when a tensile force is applied to a thin sheet with a slit pattern, three-dimensional corrugated surfaces are induced by the out-of-plane buckling. We aim to apply buckling-induced kirigami to architectural- scale deployable structures by applying a rigid origami model, i. e., a kinematic model in which rigid bodies are connected by rotating hinges, to such kirigami systems. We propose a parametric family of kirigami structures where two crease lines are added to kirigami connections. We show that the structures form one-DOF mechanisms by showing the kinematic equivalence to a degree-4 single vertex rigid origami. Also, we show that the structure within a specific parameter range can expand in two principal directions, and thus macroscopically forms an auxetic (Poisson's ratio becomes negative) material through an analysis using singular value decomposition. Furthermore, we propose a family of auxetic cellular materials obtained by assembling these structures in multiple layers. Then, we show some results in experiments on self-folding. Finally, we propose a method of using thick panels for this structure.