Minimum s - t cut in undirected planar graphs when the source and the sink are close

Consider the minimum s-t cut problem in an embedded undirected planar graph. Let p be the minimum number of faces that a curve from s to t passes through. If p = 1, that is, the vertices s and t are on the boundary of the same face, then the minimum cut can be found in O(n) time. For general planar graphs this cut can be found in O(n log n) time. We unify these results and give an O(n log p) time algorithm. We use cut-cycles to obtain the value of the minimum cut, and study the structure of these cycles to get an efficient algorithm.