Graph Theory

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Graph Theory

• Chapter 6 Connectivity and Flow

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Contents

• 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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6.1 Edge Cuts
Definition 6.1
Remark 6.2
Lemma 6.5
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Se4, e9 is an edge cut.
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6.2 Edge Connectivity and Connectivity
Definition 6.11
Remark 6.12
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Se4, e9 is an edge cut.
k'(G) ? 2
G has no bridges ? k'(G) ? 2
? k'(G) 2
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Definition 6.14
Example 6.15
k(G1) 1
k'(G1) 1
k(G2) 1
k'(G2) 2
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Example 6.17
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Exercise
1. Determine k(G) and k(G) for the following
graph.
v10
v1
v5
v6
v9
v2
v4
v8
v3
v7
2. Determine k(Km,n) and k(Km,n), where 1?m?n.
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Definition 6.19
Theorem 6.20
Note 6.21
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6.3 Blocks in Separable Graphs
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
Find the block cut-point graph for the following
graph.
v13
v14
v1
v5
v6
v2
v11
v12
v10
v4
v7
v3
v9
v8
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6.4 Flows in Networks
Definition 6.35
Definition 6.36
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Example 6.38
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6.5 The Theorems of Menger
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
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Mengers 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
Gs vertices.
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Mengers 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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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 Gs vertices.
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).