Deenition 1 given a Graph G = ( V E ) and a Subset S of V , the Cut (s) Induced

1 Minimum Cuts In this lecture we will describe an algorithm that computes the minimum cut (or simply mincut) in an undirected graph. A c u t is deened as follows. by S is the subset of edges (ii j) 2 E such that jfii jg \ Sj = 1. That is, (S) consists of all those edges with exactly one endpoint in S. Given an undirected graph G = (V E) and for each edge e 2 E a nonnegative cost (or capacity) c e , the cost of a cut (S), is the sum of the costs of the edges in the cut, that is X c((S)) = c e : e2(S) The minimum cut problem (or mincut problem) is to nd a cut of minimum cost. If all costs are 1 then the problem becomes the problem of nding a cut with as few edges as possible. Cuts are often deened in a diierent, not completely equivalent, way. Deene a cutset to b e a set of edges whose removal disconnects the graph into at least two connected components. Minimal cutsets (a minimal cutset C is a cutset such that any proper subset of C is not anymore a cutset) can be seen to correspond to cuts (S) for which the subgraphs induced by S and V ; S are connected. Observe that only minimal cutsets can be of minimum cost (among all cutsets) a n d t h a t o n l y c u t s (S) for which both S and V ; S induce connected components can b e of minimum cost (among all cuts) since the costs are assumed to be nonnegative. For this reason, the problem of nding a cutset of minimum cost is equivalent to the problem of nding a cut (S) of minimum cost, namely the mincut problem. From now o n , w e will only look at cuts (S) (and not cutsets). An important variant of the mincut problem is often considered. This is the problem of nding the minimum cost cut separating two given two vertices s and t. A cut (S) is said to separate s and t if only one of them belongs to S. We refer to this problem as the minimum (ss t)-cut problem. The minimum (ss t)-cut problem has been traditionally solved by means of network ow algorithms. Indeed it …