Menger's theorem

From Wikipedia, the free encyclopedia

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 for 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. Let G be a finite undirected graph and x and y two nonadjacent 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.

The vertex-connectivity statement of Menger's theorem is this: 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.

Menger's theorem has also been proved to hold for infinite graphs (originally a conjecture proposed by Paul Erdős): Let A and B be sets of vertices in a (possibly finite) 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.