  In two seminal papers Kontsevich used a construction called_graph homology_
as a bridge between certain infinite dimensional Lie algebras and various
topological objects, including moduli spaces of curves, the group of outer
automorphisms of a free group, and invariants of odd dimensional manifolds. In
this paper, we show that Kontsevich's graph complexes, which include graph
complexes studied earlier by Culler and Vogtmann and by Penner, have a rich
algebraic structure. We define a Lie bracket and cobracket on graph complexes,
and in fact show that they are Batalin-Vilkovisky algebras, and therefore
Gerstenhaber algebras. We also find natural subcomplexes on which the bracket
and cobracket are compatible as a Lie bialgebra.
  Kontsevich's graph complex construction was generalized to the context of
operads by Ginzburg and Kapranov, with later generalizations by
Getzler-Kapranov and Markl. In [CoV], we show that Kontsevich's results in fact
extend to general cyclic operads. For some operads, including the examples
associated to moduli space and outer automorphism groups of free groups, the
subcomplex on which we have a Lie bi-algebra structure is quasi-isomorphic to
the entire connected graph complex. In the present paper we show that all of
the new algebraic operations canonically vanish when the homology functor is
applied, and we expect that the resulting constraints will be useful in
studying the homology of the mapping class group, finite type manifold
invariants and the homology of Out(F_n).
