An extension of the Bollobás-Riordan polynomial for vertex partitioned ribbon graphs: definition and universality

In this paper we are interested in vertex partitioned ribbon graphs, which are a generalization of ribbon graphs that are studied in some theoretical physics models. We define a Hopf algebra of vertex partitioned ribbon graphs, then go on to describe how a natural generalization of the Bollobás-Riordan polynomial arises from this Hopf algebra. Using some appropriate Hopf algebraic characters we also prove the universality of our polynomial