Chapter 6 Connectivity and Flow 大葉大學 資訊工程系 黃鈴玲

 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 2

3 Definition 6.1 Remark 6.2 Lemma 6.5

4 S={e 4, e 9 } is an edge cut.

5 Definition 6.11 Remark 6.12

6 S={e 4, e 9 } is an edge cut.  '(G)  2 G has no bridges   '(G)  2   '(G) = 2

7 Definition 6.14 Example 6.15  (G 1 ) = 1  '(G 1 ) = 1  (G 2 ) = 1  '(G 2 ) = 2

8 Example 6.17

9 v1v1 v2v2 v3v3 v7v7 v4v4 v5v5 v6v6 Exercise 1. Determine  ( G ) and  ’ ( G ) for the following graph. 2. Determine  ( K m,n ) and  ’ ( K m,n ), where 1  m  n. v8v8 v9v9 v 10

10 Definition 6.19 Theorem 6.20 Note 6.21

11 Definition 6.23

12 Lemma 6.27 Definition 6.29 (Block-cutpoint graph) 

13 Definition 6.29 Corollary 6.32 Theorem 6.33

14 v1v1 v2v2 v3v3 v7v7 v4v4 v5v5 v6v6 Exercise Find the block cut-point graph for the following graph. v 10 v 14 v 13 v8v8 v9v9 v 11 v 12

15 Definition 6.35 Definition 6.36

16 Example 6.38 ( 鱈魚 )

17 > Val(f)=3500

18 Definition: u, v  V(G), Q 1 : u,v- path, Q 2 : u,v- path Q 1, Q 2 are edge-disjoint if E ( Q 1 )  E ( Q 2 ) = , Q 1, Q 2 are (internally) vertex disjoint if V ( Q 1 )  V ( Q 2 ) = { 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.

19 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.

20 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.

21 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 ). 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.