Graph_Theory_Notes4.pdf

# By convention a trivial graph is both 0 edge

• Notes
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By convention, a trivial graph is both 0-edge-connected and 1-edge- connected, but is not k -edge-connected for any k > 1. The edge connectivity of G , denoted by λ ( G ), is the maximum value of k for which G is k -edge-connected. Thus, for any nontrivial graph G , λ ( G ) = min { p 0 ( x, y ) : x, y V ( G ) , x 6 = y } . Edge cuts Definition 10. Let X be a proper subset of V ( G ). The set of edges with one end in X and the other end in V ( G ) - X is called an edge cut of G associated with X and is denoted by ( X ). For distinct x, y V ( G ), an edge cut ( X ) is said to separate x and y if x X and y V ( G ) - X . Define c 0 ( x, y ) to be the minimum cardinality of such an edge cut. Disconnecting sets In [CLZ] (and many other books), an edge cut of G is defined as a subset of E ( G ) whose removal disconnects the graph. This notion is often called a disconnecting set . It can be proved that any edge cut is a disconnecting set, but the converse is not true. However, a minimal disconnecting set must be an edge cut. Exercise: Prove these statements. Menger’s Theorem (edge version) Theorem 3. For any graph G and any two distinct vertices x and y of G , p 0 ( x, y ) = c 0 ( x, y ) . Proof. A proof will be given in Part 5 using Max-Flow-Min-Cut Theorem. Corollary 2. For any non-complete graph G , λ ( G ) = min { c 0 ( x, y ) : x, y V ( G ) , x 6 = y } . In other words, λ ( G ) is equal to the minimum cardinality of an edge cut of G .

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Examples Example 2. λ ( G ) = 0 iff G = K 1 or G is disconnected. λ ( G ) = 1 iff G is connected and G contains a bridge λ ( K n ) = n - 1 for any n 1 ( Exercise ) λ ( C n ) = 2 λ ( T ) = 1 for any tree T with at least one edge 5 Whitney’s inequalities Whitney’s inequalities Theorem 4. For any graph G , κ ( G ) λ ( G ) δ ( G ) . Proof. If G = K n is a complete graph, then κ ( G ) = λ ( G ) = δ ( G ) = n - 1 and the result is true. (Note that the trivial graph K 1 is covered here.) Assume that G is not a complete graph in the sequel.
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• One '14
• Graph Theory, Planar graph, Menger's theorem, Symmetric graph

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