1. Chapter 9 Connectivity 连通度

2. 9.1 Connectivity Consider the following graphs:  G 1 : Deleting any edge makes it disconnected.  G 2 : Cannot be disconnected by deletion of any edge; can be disconnected by deleting its cut vertex;  Intuitively, G 2 is more connected than G 1, G 3 is more connected thant G 2, and G 4 is the most connected one.

3. 9.1 Cut edges and cut vertices A cut edge of G is an edge such that G-e has more components that G. Theorem 9.1 Let G be a connected graph. The following are equivalent: 1.An edge e of G is a cut edge 2.e is not contained in any cycle of G. 3.There are two vertices u and w such that e is on every path connecting u and w.

4. 9.1 Cut edges and cut vertices Let G be a nontrivial and loopless graph. A vertex v of G is a cut vertex if G-v has more components than G. Theorem 9.2 Let G be a connected graph. The following propositions are equivalent: 1. A vertex v is a cut vertex of G 2. There are two distinct vertices u and w such that every path between u and w passes v; 3. The vertices of G can be partitioned into two disjoint vertex sets U and W such that every path between u  U and w  W passes v.

5. 9.1 Vertex cut and connectivity A vertex cut of G is a subset V’ of V such that G- V’ is disconnected. The connectivity,  (G), is the smallest number of vertices in any vertex cut of G.  A complete graph has no vertex cut. Define  (K n )=n-1;  For disconnected graph G, define  (G) = 0;  G is said to be k-connected if  (G)  k;  It is easy to see that all nontrivial connected graphs are 1-connected.   (G)=1 if and only if G=K 2 or G has a cut vertex.

6. 9.1 Edge cut and edge connectivity Let [S,S’] denote the set of edges with one end in S and the other end in S’. Let G be graph on n  2 vertices. An edge cut is a subset E’ of E(G) of the form [S,S’], where S is a nonempty proper set of V and S’=V-S. If G is nontrivial and E’ is an edge cut of G, then G- E’ is disconnected. The edge connectivity, (G), is the smallest number of edges in any edge cut.

7. 9.1 Edge cut and edge connectivity  For trivial and disconnected graph G, define (G)=0;  G is said to be k-edge-connected if (G)  k ;  All nontrivial connected graphs are 1-edge- connected. Theorem 9.3 For any connected graph G  (G)  (G)  (G) where  (G) is the smallest vertex degree of G.

8. 9.2 Menger’s theorem Theorem A graph G is k-edge-connected if and only if any two distinct vertices of G are connected by at least k edge-disjoint paths. Proof ： If there are two vertices which are connected by less than k edge-disjoint paths, then G is not k-edge- connected. On the other hand, if G is not k-edge- connected, there are edge cut that contains less than k edges, hence there are two vertices which are connected by less than k edge-disjoint paths. Theorem A graph with n  k+1 is k-connected if and only if any two distinct vertices of G are connected by at least k vertex-disjoint paths.

9. 9.3 Reliable communication networks A graph representing a communication network, the connectivity (or edge-connectivity) becomes the smallest number of stations (or links) whose breakdown would jeopardise the system. The higher the connectivity and edge connectivity, the more reliable the network. Let k be a given positive integer and let G be a weighted graph. Determine a minimum-weight k-connected spanning subgraph of G. For k=1, this is solved by Kruskal’s algorithm, for example. For k>1, the problem is unsolved.