  We present a theorem on the compatibility upon deployment of kirigami
tessellations restricted on a spherical surface with patterned slits forming
freeform quadrilateral meshes. We show that the spherical kirigami
tessellations have either one or two compatible states, i.e., there are at most
two isolated strain-free configurations along the deployment path. The theorem
further reveals that the rigid-to-floppy transition from spherical to planar
kirigami tessellations is possible if and only if the slits form parallelogram
voids along with vanishing Gaussian curvature, which is also confirmed by an
energy analysis and simulations. On the application side, we show a design of
bistable spherical dome-like structure based on the theorem. Our study provides
new insights into the rational design of morphable structures based on
Euclidean and non-Euclidean geometries.
