Menger's Theorems and Max-Flow-Min-Cut


There are several versions of Menger's Theorem, all can be derived from the Max-Flow-Min-Cut Theorem.
Undirected, Vertex Version. Let G be an undirected graph, and let u and v be nonadjacent vertices in G. Then, the maximum number of pairwise-internally-disjoint (u,v)-paths in G equals the minimum number of vertices from V(G)-{u,v} whose deletion separates u and v.
Directed, Vertex Version. Let D be a directed graph, and let u and v be vertices in G, with no arc from u to v. Then, the maximum number of pairwise-internally-disjoint directed (u,v)-paths in D equals the minimum number of vertices from V(D)-{u,v} whose deletion destroys all directed paths from u to v.
Undirected, Edge Version. Let G be an undirected graph, and let u and v be vertices in G. Then, the maximum number of pairwise-edge-disjoint (u,v)-paths in G equals the minimum number of edges from E(G) whose deletion separates u and v.
Directed, Arc Version. Let D be a directed graph, and let u and v be vertices in D. Then, the maximum number of pairwise-arc-disjoint directed (u,v)-paths in D equals the minimum number of arcs in A(D) whose deletion destroys all directed paths from u to v.
Connectivity. Let G be a k-connected graph. Then, for any pair of vertices, u and v, there are at least k pairwise-internally-disjoint (u,v)-paths in D.

Obviously, if some set of vertices, Y, separates u and v (that is, u and v are in different components of G-Y), then G is at most |Y|-connected.

Similar statements can be made for edge-connectivity, and for directed versions.
Max-Flow-Min-Cut. Let D be a directed graph, and let u and v be vertices in D. The maximum weight among all (u,v)-flows in D equals the minimum capacity among all sets of arcs in A(D) whose deletion destroys all directed paths from u to v.

Furthermore, there is a low-order polynomial-time algorithm which will find a maximum (u,v)-flow and a minimum capacity (u,v)-cut (a set of arcs in A(D) whose deletion destroys all directed paths from u to v).

Proof. We sketch the proof for the case in which all capacities are +1. A {0,1}-flow is function from the set of arcs to the set {0,1), such that sum of the flow on arcs into any vertex is the same as the sum of the flow on arcs out of the vertex, except for the two special vertices: the source, u, and the sink, v. The 0-flow is the flow which is zero on every arc.
Starting with any {0,1}-flow f, for example the 0-flow, we make a new directed graph, D', with V(D')=V(D) and with A(D') = A1 union A2, where A1 is the set of those arcs of D whose flow is zero, and A2 is the set of reversals of those arcs of D whose flow is one.

Suppose that there is a directed (u,v)-path, P, in D'. Let F denote the symmetric difference of A1 and P. Clearly, F is the set of edges of a flow f', where f' is one on arcs of F and zero on arcs not in F. Furthermore, F contains more arcs which are incident with u than A1 does.

Suppose, instead, that there is no directed (u,v)-path in D'. Let S denote the set of vertices in D' to which there is a directed path from u in D'. Then, in D, the set of arcs, C, leaving S must have cardinality exactly the same as the cardinality of the sets of arcs from A1 which are incident with u.

The sum of the capacities on the arcs of A1 which are incident with u is usually called the weight of the flow. Obviously, no {0,1}-flow can have weight exceeding |C|.
Last modified July 3, 1996, by S.C. Locke. How to contact me.