The first instances of graph complexes have been introduced by
Kontsevich as a way to expose highly non-trivial
interrelations between certain infinite dimensional Lie algebras
and topological objects, including moduli spaces of curves,
invariants of odd dimensional manifolds, and the group of outer
automorphisms of a free group.
We develop a new technique for computing cohomology of an important
class of directed graph complexes with wheels in terms of
other much simpler purely operadic graph complexes.
We apply this technique to compute cohomology of several
classical examples and, as an application, give a new
proof of celebrated Kontsevich's theorem on existence of star
products on formal germs of Poisson manifolds.