We investigate combinatorial and algebraic aspects of the interplay between
renormalization and monodromies for Feynman amplitudes. We clarify how
extraction of subgraphs from a Feynman graph interacts with putting edges
onshell or with contracting them to obtain reduced graphs. Graph by graph this
leads to a study of cointeracting bialgebras. One bialgebra comes from
extraction of subgraphs and hence is needed for renormalization. The other
bialgebra is an incidence bialgebra for edges put either on- or offshell. It is
hence related to the monodromies of the multivalued function to which a
renormalized graph evaluates. Summing over infinite series of graphs,
consequences for Green functions are derived using combinatorial
Dyson--Schwinger equations.Comment: 76 pages, a few extra remarks and more reference