In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices.
Proved by Karl Menger in 1927, it characterizes the connectivity of a graph.
It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs.
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Edge connectivity
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 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-disjoint paths from x to y.
The implication for the graph G is the following version:
A graph is k-edge-connected (it remains connected after removing fewer than k edges) if and only if every pair of vertices has k edge-disjoint paths in between.
Vertex connectivity
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 size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally disjoint paths from x to y.
A consequence for the entire graph G is this version:
A graph is k-vertex-connected (it has more than k vertices and it remains connected after removing fewer than k vertices) if and only if every pair of vertices has at least k internally disjoint paths in between.
Directed graphs
All these statements in both edge and vertex versions remain true in directed graphs (when considering directed paths).
Short proof
Most direct proofs consider a more general statement to allow proving it by induction. It is also convenient to use definitions that include some degenerate cases.
The following proof for undirected graphs works without change for directed graphs or multi-graphs, provided we take path to mean directed path.
For sets of vertices A,B ⊂ G (not necessarily disjoint), an AB-path is a path in G with a starting vertex in A, a final vertex in B, and no internal vertices either in A or in B. We allow a path with a single vertex in A ∩ B and zero edges.
An AB-separator of size k is a set S of k vertices (which may intersect A and B) such that G−S contains no AB-path.
An AB-connector of size k is a union of k vertex-disjoint AB-paths.
Theorem: The minimum size of an AB-separator is equal to the maximum size of an AB-connector.
In other words, if no k−1 vertices disconnect A from B, then there exist k disjoint paths from A to B.
This variant implies the above vertex-connectivity statement: for x,y ∈ G in the previous section, apply the current theorem to G−{x,y} with A = N(x), B = N(y), the neighboring vertices of x,y. Then a set of vertices disconnecting x and y is the same thing as an
AB-separator, and removing the end vertices in a set of independent xy-paths gives an AB-connector.
Proof of the Theorem:
Induction on the number of edges in G.
For G with no edges, the minimum AB-separator is A ∩ B,
Other proofs
The directed edge version of the theorem easily implies the other versions.
To infer the directed graph vertex version, it suffices to split each vertex v into two vertices v1, v2, with all ingoing edges going to v1, all outgoing edges going from v2, and an additional edge from v1 to v2.
The directed versions of the theorem immediately imply undirected versions: it suffices to replace each edge of an undirected graph with a pair of directed edges (a digon).
The directed edge version in turn follows from its weighted variant, the max-flow min-cut theorem.
Its proofs are often correctness proofs for max flow algorithms.
It is also a special case of the still more general (strong) duality theorem for linear programs.
A formulation that for finite digraphs is equivalent to the above formulation is:
Let A and B be sets of vertices in a finite digraph G. Then there exists a family P of disjoint AB-paths and an AB-separating set that consists of exactly one vertex from each path in P.
In this version the theorem follows fairly easily from Kőnig's theorem: in a bipartite graph, the minimal size of a cover is equal to the maximal size of a matching.
This is done as follows: replace every vertex v in the original digraph D by two vertices v', v'', and every edge uv by the edge u'v''; additionally, include the edges v'v'' for every vertex v that is neither in A nor B. This results in a bipartite graph, whose one side consists of the vertices v', and the other of the vertices v''.
Infinite graphs
Menger's theorem holds for infinite graphs, and in that context it applies to the minimum cut between any two elements that are either vertices or ends of the graph (Halin 1974). The following result of Ron Aharoni and Eli Berger was originally a conjecture proposed by Paul Erdős, and before being proved was known as the Erdős–Menger conjecture.
It is equivalent to Menger's theorem when the graph is finite.
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.