  We introduce a new method to generate duality relations for correlation
functions of the Potts model on planar graphs. The method extends previously
known results, by allowing the consideration of the correlation function for
arbitrarily placed vertices on the graph. We show that generally it is linear
combinations of correlation functions, not the individual correlations, that
are related by dualities. The method is illustrated in several non-trivial
cases, and the relation to earlier results is explained. A graph-theoretical
formulation of our results in terms of rooted dichromatic, or Tutte,
polynomials is also given.
