A Coarse Tutte Polynomial for Hypermaps

We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction reductions where the terminal forms consist of isolated vertices. Our Tutte polynomial extends the classical Tutte polynomial of a graph as well as the Tutte polynomial of an embedded graph (i.e., the ribbon graph polynomial), and it is a specialization of the transition polynomial via a medial map transformation. We give hypermap duality and partial duality identities for our polynomial, as well as some evaluations, and examine relations between our polynomial and other hypermap polynomials.