Menger's Theorem

In the mathematical discipline of graph theory and related areas, Menger's theorem is a basic result about connectivity in finite undirected graphs. It was proved for edge-connectivity and vertex-connectivity by Karl Menger in 1927. The edge-connectivity version of Menger's theorem was later generalized by the max-flow min-cut theorem.

The edge-connectivity version of Menger's theorem is as follows:

Let G be a finite undirected graph and x and y two distinct vertices. Then the theorem states that the size of the minimum edge cut for x and y (the minimum number of edges whose removal disconnects x and y) is equal to the maximum number of pairwise edge-independent paths from x to y.

Extended to subgraphs: a maximal subgraph disconnected by no less than a k-edge cut is identical to a maximal subgraph with a minimum number k of edge-independent paths between any x, y pairs of nodes in the subgraph.

The vertex-connectivity statement of Menger's theorem is as follows:

Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the theorem states that the size of the minimum vertex cut for x and y (the minimum number of vertices whose removal disconnects x and y) is equal to the maximum number of pairwise vertex-independent paths from x to y.

Extended to subgraphs: a maximal subgraph disconnected by no less than a k-vertex cut is identical to a maximal subgraph with a minimum number k of vertex-independent paths between any x, y pairs of nodes in t is equivalent to Menger's theorem for finite graphs and is a deep recent result of Ron Aharoni and Eli Berger for infinite graphs (originally a conjecture proposed by Paul Erdős):

Let A and B be sets of vertices in a (possibly infinite) digraph G. Then there exists a family P of disjoint A-B-paths and a separating set which consists of exactly one vertex from each path in P.