  In this paper, we introduce the notion of bigraft algebra, generalizing the
notions of left and right graft algebras. We give a combinatorial description
of the free bigraft algebra generated by one generator and we endow this
algebra with a Hopf algebra structure, and a pairing. Next, we study the Koszul
dual of the bigraft operad and we give a combinatorial description of the free
dual bigraft algebra generated by one generator. With the help of a rewriting
method, we prove that the bigraft operad is Koszul. Finally, we define the
notion of infinitesimal bigraft bialgebra and we prove a rigidity theorem for
connected infinitesimal bigraft bialgebras.
