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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.
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.
The connectivity and edge-connectivity of G can then be computed as the minimum values of κ(u, v) and λ(u, v), respectively. In computational complexity theory , SL is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to L by Omer Reingold in ...
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 ...
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.
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 .
Karl Menger was a young geometry professor at the University of Vienna and Arthur Cayley was a British mathematician who specialized in algebraic geometry. Menger extended Cayley's algebraic results to propose a new axiom of metric spaces using the concepts of distance geometry up to congruence equivalence, known as the Cayley–Menger determinant.
The strong connectivity augmentation problem was formulated by Kapali Eswaran and Robert Tarjan . They showed that a weighted version of the problem is NP-complete, but the unweighted problem can be solved in linear time. [1] Subsequent research has considered the approximation ratio and parameterized complexity of the weighted problem. [2] [3]