  We discuss a graph complex formed by directed acyclic graphs with external
legs. This complex comes in particular with a map to the ribbon graph complex
computing the (compactly supported) cohomology of the moduli space of points
$\mathcal M_{g,n}$, extending an earlier result of Merkulov-Willwacher. It is
furthermore quasi-isomorphic to the hairy graph complex computing the weight 0
part of the compactly supported cohomology of $\mathcal{M}_{g,n}$ according to
Chan-Galatius-Payne. Hence we can naturally connect the works
Chan-Galatius-Payne and of Merkulov-Willwacher and the ribbon graph complex and
obtain a fairly satisfying picture of how all the pieces and various graph
complexes fit together, at least in weight zero.
