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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 ...
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 .
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.
Robbins (1939) introduces the problem of strong orientation with a story about a town, whose streets and intersections are represented by the given graph G.According to Robbins' story, the people of the town want to be able to repair any segment of road during the weekdays, while still allowing any part of the town to be reached from any other part using the remaining roads as two-way streets.
The concept of a gammoid was introduced and shown to be a matroid by Hazel Perfect , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths. [1] Gammoids were given their name by Pym (1969) [2] and studied in more detail by Mason (1972). [3]