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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 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-disjoint paths from x to y.
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.
By Menger's theorem, for any two vertices u and v in a connected graph G, the numbers κ(u, v) and λ(u, v) can be determined efficiently using the max-flow min-cut algorithm. The connectivity and edge-connectivity of G can then be computed as the minimum values of κ(u, v) and λ(u, v), respectively.
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 ...
In the undirected edge-disjoint paths problem, we are given an undirected graph G = (V, E) and two vertices s and t, and we have to find the maximum number of edge-disjoint s-t paths in G. Menger's theorem states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
In this case () is the maximum number of edge-disjoint s-t paths, and () is the size of the smallest edge-cut separating s and t, so Menger's theorem (edge-connectivity version) asserts that () = (). Let G be a connected graph and let H be the clutter on E ( G ) {\displaystyle E(G)} consisting of all edge sets of spanning trees of G .
As mentioned above, the k = 2 case of the Lovász–Woodall conjecture follows from Menger's theorem. The k = 3 case was given as an exercise by Lovász. [7] After the conjecture was made, it was proven for k = 4 by Péter L. Erdős and E. Győri [8] and independently by Michael V. Lomonosov., [9] and for k = 5 by Daniel P. Sanders.
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