  We introduce WI-posets as intermediate objects in the study of Z_2-homotopy
types of graph complexes. It turns out that (almost) all graph complexes
associated to a graph can be viewed as avatars of the same object, as long as
their Z_2-homotopy types are concerned. Among the applications are a proof that
each finite, free Z_2-complex is a graph complex and an evaluation of
Z_2-homotopy types of complexes Ind(C_n) of independence sets in a cycle C_n.
The main tools used in the paper are Quillen fiber theorem and Bredon criterion
for Z_2-equivalence of Z_2-complexes.
