Menger's theorem

In the mathematical discipline of graph theory and related areas, Menger's theorem is a characterization of the connectivity in finite undirected graphs in terms of the minimum number of disjoint paths that can be found between any pair of vertices. 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:

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

The directed graphs vertex version of the theorem implies the undirected version by a familiar device of replacing every edge by a digon (doubly directed edge). The validity of the theorem for the resulting digraph easily implies its validity for the original, undirected, graph.

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 A-B-paths and an A-B separating set that consists of exactly one vertex from each path in P.

In this version the theorem follows in a straightforward manner from König's duality 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''. This results in a bipartite graph, whose one side consists of the vertices v', and the other of the vertices v''.

Applying König's theorem we obtain a cover C and a matching M of the same size. Add to C all vertices a'', a in A, and all vertices b', b in B. Let now the Q in the original graph consist of all vertices v such that both v' and v'' belong to C, and let P be the set of all A-B paths in D such that x'y'' belongs to M for every xy in the edge set of the path. It is straightforward to check that Q is an A-B separating set, and that every path in the family P contains precisely one vertex from Q, as desired.

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