Graph complexes and Deformation theories of the (wheeled) properads of quasi- and pseudo-Lie bialgebras

Quasi-Lie bialgebras are natural extensions of Lie-bialgebras, where the cobracket satisfies the co-Jacobi relation up to some natural obstruction controlled by a skew-symmetric 3-tensor $\phi$. This structure was introduced by Drinfeld while studying deformation theory of universal enveloping algebras and has since seen many other applications in algebra and geometry. In this paper we study the derivation complex of strongly homotopy quasi-Lie bialgebra, both in the unwheeled (i.e standard) and wheeled case, and compute its cohomology in terms of Kontsevich graph complexes.