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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.
More generally, an edge cut of G is a set of edges whose removal renders the graph disconnected. The edge-connectivity λ(G) is the size of a smallest edge cut, and the local edge-connectivity λ(u, v) of two vertices u, v is the size of a smallest edge cut disconnecting u from v. Again, local edge-connectivity is symmetric.
The edge connectivity of is the maximum value k such that G is k-edge-connected. The smallest set X whose removal disconnects G is a minimum cut in G . The edge connectivity version of Menger's theorem provides an alternative and equivalent characterization, in terms of edge-disjoint paths in the graph.
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The vertex-connectivity of an input graph G can be computed in polynomial time in the following way [4] consider all possible pairs (,) of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for (,) is the number of pairwise vertex-independent paths between them, encode the input by doubling each vertex as an edge to reduce to a computation of the ...
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This gives immediately a linear-time 2-connectivity test and can be extended to list all cut vertices of G in linear time using the following statement: A vertex v in a connected graph G (with minimum degree 2) is a cut vertex if and only if v is incident to a bridge or v is the first vertex of a cycle in C – C 1. The list of cut vertices can ...