The Kontsevich graph orientation morphism revisited

The orientation morphism $Or(\cdot )(P)\text{\hspace{0.17em}⁣}:\gamma \mapsto \dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ with ordered sets of edges and without multiple edges and one-cycles. It is known that $d$-cocycles $\mathit{\gamma }\in \ker d$ with respect to the vertex-expanding differential $d=[\bullet \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\bullet ,\cdot ]$ are mapped by $Or$ to Poisson cocycles $Q(P)\in \ker [\text{\hspace{0.17em}⁣}[P,\cdot ]\text{\hspace{0.17em}⁣}]$, that is, to infinitesimal symmetries of Poisson bi-vectors $P$. The formula of orientation morphism $Or$ was expressed in terms of the edge orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle $\mathrm{\Pi }{T}^{\ast }{N}^d$ over any $d$-dimensional affine real Poisson manifold $N^d$. We express this formula in terms of (un)oriented graphs themselves, i.e. without explicit reference to supermathematics on $\mathrm{\Pi }{T}^{\ast }{N}^d$.