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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.
==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 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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