  We introduce and study the notion of a dual Feynman transform of a modular
operad. This generalizes and gives a conceptual explanation of Kontsevich's
dual construction producing graph cohomology classes from a contractible
differential graded Frobenius algebra. The dual Feynman transform of a modular
operad is indeed linear dual to the Feynman transform introduced by Getzler and
Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman
transform, the dual notion admits an extremely simple presentation via
generators and relations; this leads to an explicit and easy description of its
algebras. We discuss a further generalization of the dual Feynman transform
whose algebras are not necessarily contractible. This naturally gives rise to a
two-colored graph complex analogous to the Boardman-Vogt topological tree
complex.
