chapter 6 connectivity and flow 大葉大學 資訊工程系 黃鈴玲 2010.11
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Chapter 6 Connectivity and Flow
大葉大學 資訊工程系 黃鈴玲2010.11
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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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v1
v2
v3 v7
v4
v5 v6
Exercise Exercise
1. Determine (G) and ’(G) for the following graph.
2. Determine (Km,n) and ’(Km,n), where 1mn.
v8
v9
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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v1
v2
v3v7
v4
v5 v6
Exercise Exercise
Find the block cut-point graph for the following graph.
v10
v14v13
v8
v9
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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<< 以下改用另一份投影片 , 舊版 ch5>>
1600
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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.