In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra structure on the deformation complex of the associative algebra of functions on $\mathbb R^n$ (called $\mathcal D_{\rm poly}$) and its cohomology the differential graded algebra of polyvector fields (called $\mathcal T_{\rm poly}$). The explicit construction involves a sum $\mathcal U_n(X_1,\ldots,X_n)=\sum_{\Gamma}\omega_\Gamma B_\Gamma(X_1,\ldots,X_n)$ over a particular class of graphs $\Gamma$ (called admissible graphs) where $\omega$ is a map from these graphs to ℝ (the weight function) and $B$ maps a graph and a set of compatible polyvector fields to a multidifferential operator (i.e. an element of $\mathcal D_{\rm poly}$).

The graphs are characterised by two integers $n$ and $m$ standing for the number of aerial vertices (decorated by polyvector fields via the action of $B$) and terrestrial vertices (decorated by functions) respectively.

Imposing that the component $\mathcal U_n$ of the morphism $\mathcal U$ is of degree $1-n$ imposes the following constraint on the number of edges of admissible graphs: e=2n+m-2 which coincides with the dimension of configuration spaces over which the integral $\omega_\Gamma$ is integrated.

**Questions**:

1) Is there a natural operad structure on the set of admissible graphs ? (I know there is a large body of literature regarding operad structure on the Kontsevich graph complex introduced in [Formality conjecture] but which involves different (linear combinations of isomorphisms classes of) graphs).

2) More generally, is there a way to interpret the constraint e=2n+m-2 purely in terms of a certain algebraic structure on the set of admissible graphs (without referring to $\mathcal U$) ?