Definitions

Menger

Menger

[meng-er]
Menger, Carl, 1840-1921, Austrian economist, a founder of the Austrian school of economics. He was professor of economics at the Univ. of Vienna from 1873 until 1903, when he retired to devote himself to research. Following an empirical approach rather than the historical method, he formulated a theory of marginal utility. The basic principle is that consumer goods have value of two orders, as they serve human needs directly or indirectly; thus he explained the economic phenomena of price and distribution in terms of social value. His theories are well known to the English-speaking world through the works of some of his associates, especially Friedrich von Wieser and Eugen von Böhm-Bawerk. In response to a particularly negative review of Menger's Problems of Economics and Sociology (1883) by Gustav Schmoller, Menger published a critique of the historical school of economics. This exchange resulted in long-standing animosity between the two schools of economic thought. His chief work is Principles of Economics (1871; tr. 1950).
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 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:

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.

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 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.

Extended to subgraphs: a maximal subgraph disconnected by no less than a k-vertex cut is identical to a maximal subgraph with a minimum number k of vertex-independent paths between any x, y pairs of nodes in the subgraph.

It is not too hard to show that Menger's theorem holds for infinite graphs. The following statement is equivalent to Menger's theorem for finite graphs and is a deep recent result of Ron Aharoni and Eli Berger for infinite graphs (originally a conjecture proposed by Paul Erd%C5%91s): 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.

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