By Berge C.

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Fulkerson, Maximal flow through a network, Canad. J. Math. 8 (1956), 399–404. 25. G. Fricke, O. R. Oellermann and H. C. Swart, The average edge-connectivity and degree conditions, preprint. 26. T. Gallai, Maximum-minimum Sätze and verallgemeinerte Faktoren von Graphen, Acta Math. Acad. Sci. Hungar. 12 (1961), 131–173. 27. D. L. Goldsmith, On the second-order edge-connectivity of a graph, Congr. Numer. 29 (1980), 479–484. 28. D. L. Goldsmith, On the nth order connectivity of a graph. Congr. Numer.

7. Connectivity of sets The beauty of Menger’s theorem lies in the way that it relates separating sets to paths joining pairs of vertices. In this section we consider extensions of these concepts to sets of more than two vertices. Our story begins with sets of paths in a graph joining (independent) sets of vertices in a connected subgraph. In Section 8 we move on to trees joining unrestricted sets of vertices. Total separation Let G be a graph, and let A be an independent set of vertices with at least two vertices.

Menger’s theorem also holds for infinite cardinals, but Erd˝os proposed a better way of extending the theorem to infinite graphs when he made the following classic conjecture. Conjecture A Let V and W be sets of vertices in an infinite graph G. Then G contains a set of disjoint V –W paths and a V –W separating set S that are in one-to-one correspondence, where each vertex of S lies on exactly one path in , and each path in contains exactly one vertex of S. Podewski and Steffens [67] made some progress on this conjecture by proving it to be true for countable graphs that contain no infinite paths.

### Hypergraphs by Berge C.

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