Special divisors in special divisor classes on graphs

We generalize the notion of divisors that are reduced with respect to a vertex to divisors that are reduced with respect to a set of vertices. We establish properties that remain valid in this more general context which allow us to improve the standard algorithm to determine whether a divisor class is effective. We then characterize reduced divisors in our sense in terms of Luo’s potential-theoretic generalization of reduced divisors. Finally, as a further application, we use this setup to study the existence of so called uniform representatives in special divisor classes.