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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 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.
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.
A cut (,) in an undirected graph = (,) is a partition of the vertices into two non-empty, disjoint sets =.The cutset of a cut consists of the edges {:,} between the two parts. . The size (or weight) of a cut in an unweighted graph is the cardinality of the cutset, i.e., the number of edges between the two parts
Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian–American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves and regions; in graph theory , he is credited with Menger's theorem .
An edge connector is the portion of a printed circuit board (PCB) consisting of traces leading to the edge of the board that are intended to plug into a matching socket. The edge connector is a money-saving device because it only requires a single discrete female connector (the male connector is formed out of the edge of the PCB), and they also ...