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.
[edit] References
- Menger, Karl (1927). "Zur allgemeinen Kurventhoerie". Fund. Math. 10: 96-115.
- Aharoni, Ron and Berger, Eli. "Menger's Theorem for infinite graphs".
