The cohomology of the full directed graph complex

In his seminal paper “Formality conjecture”, M. Kontsevich introduced a graph complex $G{C}_{1ve}$ closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex explicitly in terms of the cohomology of $G{C}_{1ve}$. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting.