An x y vertex cut is a subset s of v g x y such that

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An ( x, y ) -vertex-cut is a subset S of V ( G ) - { x, y } such that x and y are in different components of G - S . We also say that S separates x and y . Define c ( x, y ) to be the minimum cardinality of an ( x, y )-vertex-cut. (If x = y or xy E ( G ), then c ( x, y ) is not defined.) Definition 6. A subset of V ( G ) is called a vertex cut of G if it is an ( x, y )-vertex-cut for some pair of nonadjacent vertices x and y . Equivalently, a vertex cut of a non-complete graph G is a subset S of V ( G ) such that G - S is disconnected. 3 Menger and Whitney Menger’s Theorem (vertex version) Theorem 1. For any graph G and any two nonadjacent vertices x and y of G , p ( x, y ) = c ( x, y ) . Proof. A proof will be given in Part 5 using Max-Flow-Min-Cut Theorem. See [CLZ, pp.104-105] for a direct proof. Whitney’s Theorem Theorem 2. For any non-complete graph G , κ ( G ) = min { p ( x, y ) : x, y V ( G ) , x 6 = y, xy 6∈ E ( G ) } . Proof. See [J.A. Bondy and U.S.R. Murty, Graph Theory, 2008, Theorem 9.2, pp.210-211]. Corollary 1. For any non-complete graph G , κ ( G ) = min { c ( x, y ) : x, y V ( G ) , x 6 = y, xy 6∈ E ( G ) } . In other words, κ ( G ) is equal to the minimum cardinality of a vertex cut of G . Cut-vertices and graphs with connectivity 1 Definition 7. A cut-vertex of a graph G is a vertex v such that k ( G - v ) > k ( G ). Lemma 1. Every non-leaf vertex in a tree is a cut-vertex. Lemma 2. Let G be a graph. Then κ ( G ) = 1 iff G = K 1 , G = K 2 , or G is connected but contains a cut-vertex.
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4 Edge connectivity Local edge connectivity Definition 8. Let G be a graph, and let x and y be distinct vertices of G . Two ( x, y )-paths in G are edge-disjoint if they do not have common edges (but they are allowed to have common vertices). The local edge connectivity between distinct x and y , denoted by p 0 ( x, y ), is the maximum number of pairwise edge-disjoint ( x, y )-paths in G . (When x = y , p 0 ( x, y ) is undefined.) Edge connectivity Definition 9. A nontrivial graph G is called k -edge-connected if p 0 ( x, y ) k for any two distinct vertices x and y .
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