Chapter 6 Connectivity and Flow

大葉大學 資訊工程系 黃鈴玲2010.11

6.1 Edge Cuts 6.2 Edge Connectivity and Connectivity 6.3 Blocks in Separable Graphs 6.4 Flows in Networks 6.5 The Theorems of Menger

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Definition 6.1

Remark 6.2

Lemma 6.5

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S={e4, e9} is an edge cut.

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Definition 6.11

Remark 6.12

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S={e4, e9} is an edge cut.

'(G) 2

G has no bridges '(G) 2

'(G) = 2

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Definition 6.14

Example 6.15

(G1) = 1

'(G1) = 1(G2) = 1

'(G2) = 2

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Example 6.17

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v3 v7

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Exercise Exercise

1. Determine (G) and ’(G) for the following graph.

2. Determine (Km,n) and ’(Km,n), where 1mn.

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v10

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Definition 6.19

Theorem 6.20

Note 6.21

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Definition 6.23

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Lemma 6.27

Definition 6.29 (Block-cutpoint graph)

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Definition 6.29

Corollary 6.32

Theorem 6.33

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Exercise Exercise

Find the block cut-point graph for the following graph.

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v14v13

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v11 v12

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Definition 6.35

Definition 6.36

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Example 6.38 ( 鱈魚 )

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Val(f)=3500

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Definition: u, v V(G), Q1 : u,v-path, Q2: u,v-path

Q1, Q2 are edge-disjoint if E(Q1) E(Q2) = ,

Q1, Q2 are (internally) vertex disjoint if V (Q1) V(Q2) = { u, v }

Menger’s Theorem (directed edge version): Let G be a directed graph and u, v V(G). The

maximum number of edge-disjoint directed u, v-paths is equal to the minimum number of edges needed to be removed from G to destroy all u, v-paths.

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Menger’s Theorem (edge version): Let G be a graph and u, v V(G). The maximum

number of edge-disjoint u, v-paths in G is equal to the minimum number of edges needed to be removed from G to disconnect u from v.

Theorem 6.59 A connected graph G is k-edge-connected if, and

only if, there are at least k edge-disjoint paths between each pair of G’s vertices.

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Menger’s Theorem (directed vertex version): Let G be a directed graph and u, v V(G). The

maximum number of vertex-disjoint directed u, v-paths is equal to the minimum number of vertices, other than u and v, needed to be removed from G to destroy all directed u, v-paths.

Menger’s Theorem (vertex version): Let G be a graph and u, v V(G). The maximum

number of vertex-disjoint u, v-paths in G is equal to the minimum number of vertices needed to be removed from G to disconnect u from v.

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Ex1. Let G be an n-connected graph of p vertices. Show that p n (diam(G) 1) + 2. Ex1. Let G be an n-connected graph of p vertices. Show that p n (diam(G) 1) + 2.

Ex2. Let G be an n-edge-connected graph of q edges. Show that q n diam(G). Ex2. Let G be an n-edge-connected graph of q edges. Show that q n diam(G).

Theorem 6.58 A connected graph G is k-connected if, and only

if, there are at least k vertex-disjoint (excluding endvertices) paths between each pair of G’s vertices.