  We develop a bipartite rigidity theory for bipartite graphs parallel to the
classical rigidity theory for general graphs, and define for two positive
integers $k,l$ the notions of $(k,l)$-rigid and $(k,l)$-stress free bipartite
graphs. This theory coincides with the study of Babson--Novik's balanced
shifting restricted to graphs. We establish bipartite analogs of the cone,
contraction, deletion, and gluing lemmas, and apply these results to derive a
bipartite analog of the rigidity criterion for planar graphs. Our result
asserts that for a planar bipartite graph $G$ its balanced shifting, $G^b$,
does not contain $K_{3,3}$; equivalently, planar bipartite graphs are
generically $(2,2)$-stress free. We also discuss potential applications of this
theory to Jockusch's cubical lower bound conjecture and to upper bound
conjectures for embedded simplicial complexes.
