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. 1 Introduction The minimum s − t cut problem is a well-studied problems with applications in many fields. By the Max-Flow Min-Cut Theorem [4], the value of the minimum s − t cut is the same as the value of the maximum s − t flow, and a minimum cut can be easily obtained from a maximum flow. A planar graph is a graph that has an embedding in the plane such that no pair of edges cross each other. General maximum flow algorithms can solve the maximum flow and the minimum cut problems on planar graphs with n vertices and O(n) edges in O(n 2 log n) time. On the other hand, algorithms that take advantage of the structure of the planar embedding of the graph can find the minimum cut and the maximum flow in O(n log n) time (see below). The history of the maximum problem on planar graphs is surveyed in [2]. In this paper we focus on undirected planar graphs. Itai and Shiloach [11] used the correspondence between an s − t cut and a cycle in the dual planar graph (see Sect. 2) separating the dual face s * that correspond to s from the dual face t * that corresponds to t. Such a cycle is called a cut-cycle. Itai and Shiloach gave an O(n 2 log n) time algorithm for finding a minimum cut using cut-cycles in undirected planar graphs. Reif [18] improved the time bound of the algorithm to O(n log 2 n) using a divide-and-conquer approach. Frederickson [5] improved the time bound of the last algorithm to O(n log n) by providing a faster shortest paths algorithm. Hassin and Johnson [8] completed the picture by showing how to find also the …