The graph orientation morphism: constructing a symmetry by using a graph cocycle.
* Endomorphisms of the space of multivector fields. The Schouten bracket, its own Jacobi identity; the master-equation. The Richardson--Nijenhuis bracket, its own Jacobi identity; the master equation.
* The edge (the stick in the construction of graph differential). The graph orientation morphism Or takes the stick to the Schouten bracket.
* Verifying the property of the image of a graph cocycle under Or to be a Poisson cocycle. Canonical and incidental solutions of the problem of factorization via the Jacobiator. Topological identities in the spaces of Leibniz graphs.
* Lie algebra homomorphism: from graph cocycles to symmetries of Poisson brackets.