  We study the hairy graph homology of a cyclic operad; in particular we show
how to assemble corresponding hairy graph cohomology classes to form cocycles
for ordinary graph homology, as defined by Kontsevich. We identify the part of
hairy graph homology coming from graphs with cyclic fundamental group as the
dihedral homology of a related associative algebra with involution. For the
operads Comm, Assoc and Lie we compute this algebra explicitly, enabling us to
apply known results on dihedral homology to the computation of hairy graph
homology. In addition we determine the image in hairy graph homology of the
trace map defined in [CKV], as a symplectic representation.
  For the operad Lie assembling hairy graph cohomology classes yields all known
non-trivial rational homology of Out(F_n). The hairy graph homology of Lie is
also useful for constructing elements of the cokernel of the Johnson
homomomorphism of a once-punctured surface.
