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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 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 ...
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 ...
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.
An example graph, with 6 vertices, diameter 3, connectivity 1, and algebraic connectivity 0.722. The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1]
Equivalently, the connectivity of a graph is the greatest integer k for which the graph is k-connected. While terminology varies, noun forms of connectedness-related properties often include the term connectivity. Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple ...
The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not regarded as 2-connected. This property is especially useful in maintaining a graph with a two-fold redundancy , to prevent disconnection upon the removal of a single edge (or connection).