  We introduce the study of forcing sets in mathematical origami. The origami
material folds flat along straight line segments called creases, each of which
is assigned a folding direction of mountain or valley. A subset $F$ of creases
is forcing if the global folding mountain/valley assignment can be deduced from
its restriction to $F$. In this paper we focus on one particular class of
foldable patterns called Miura-ori, which divide the plane into congruent
parallelograms using horizontal lines and zig-zag vertical lines. We develop
efficient algorithms for constructing a minimum forcing set of a Miura-ori map,
and for deciding whether a given set of creases is forcing or not. We also
provide tight bounds on the size of a forcing set, establishing that the
standard mountain-valley assignment for the Miura-ori is the one that requires
the most creases in its forcing sets. Additionally, given a partial
mountain/valley assignment to a subset of creases of a Miura-ori map, we
determine whether the assignment domain can be extended to a locally
flat-foldable pattern on all the creases. At the heart of our results is a
novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of
grid graphs.
