On the Theory of Pfaffian Orientations. II. T-joins, k-cuts, and Duality of Enumeration

This is a continuation of our paper \A Theory of Pfaaan Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of T-joins and k-cuts of graphs. As a consequence, we show that the computational problem to nd the maximum weight of an edge-cut is polynomially solvable for the instances (G; w) where G is a graph embedded on an arbitrary xed orientable surface and the weight function w has only a bounded number of diierent values. We also survey the related results concerning a duality of the Tutte polynomial, and present an application for the weight enumerator of a binary code. In a continuation of this paper which is in preparation we present an application to the Ising problem of three-dimensional crystal structures.