Chapter 9 Connectivity 连通度

1. Chapter 9 Connectivity连通度

2. 9.1 Connectivity Consider the following graphs: • G1: Deleting any edge makes it disconnected. • G2: Cannot be disconnected by deletion of any edge; can be disconnected by deleting its cut vertex; • Intuitively, G2 is more connected than G1, G3 is more connected thant G2, and G4 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: • An edge e of G is a cut edge • e is not contained in any cycle of G. • 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 (Kn)=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=K2 or G has a cut vertex.

6. 9.1 Edge cut and edge connectivity Let G be graph on n2 vertices. An edge cut is a subset E’ of E(G) such that G-E’ is disconnected. The edge connectivity, (G), is the smallest number of edges in any edge cut. • 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.

7. 9.1 Edge cut and edge connectivity Find (G), (G) and (G) for the following graphs Theorem 9.3 For any connected graph G (G) (G)(G) where (G) is the smallest vertex degree of G.

8. 容易说明(G)(G)，只需将最小度数结点的关联边删除可得非连通图。如左图所示。容易说明(G)(G)，只需将最小度数结点的关联边删除可得非连通图。如左图所示。 如何说明(G) (G)？ • 设E={e1,e2,…,ek}是割边集，删除每个边ei的一个端点便删除了E，从而得到非连通图？ • E={(u,w),(v,x)}是割边集，删除u,v后图仍然是连通的。 • 假定删除E后有连通分支G1, G2, 必有v1V(G1),v2V(G2), v1,v2不相邻(见课本)，应该保留v1,v2，删除E的k个结点使得E被删除。如右图，保留u,x, 删除w,v.

9. 9.1 Edge cut and edge connectivity Theorem 9.4 (Whitney) A graph G of order n(3) is 2-connected if and only if any two vertices of G are in a common cycle. • The theorem can be proved by induction on the length of the paths. See the textbook for the proof. • State it in another way: A graph G of order n(3) is 2-connected if and only if any two vertices of G connected by at least 2 vertex-disjoint paths.

10. 9.1 Edge cut and edge connectivity Theorem 9.5 (Whitney) A graph G of order n(3) is 2-edge-connected if and only if any two vertices of G are in a simple closed path. • State it in another way: A graph G of order n(3) is 2-edge-connected if and only if any two vertices of G are connected by at least 2 edge-disjoint paths.

11. p u s x q v w y r t z 9.1 Edge-disjoint paths Let v and w are two vertices in a graph. A collection of paths from v to w are called edge-disjoint paths if no two paths in it share an edge. Count the number of edge-disjoint paths from v to w in the graph above. Find the edge connectivity of the graph.

12. p u s x q v w y r t z 9.1 Vertex-disjoint paths Similarly, we can define vertex-disjoint paths. Find the connectivity of the graph and the number of vertex-disjoint paths from v to w.

13. p u s x q v w y r t z 9.2 Menger’s theorem Theorem 9.6 The maximum number of edge-disjoint paths connecting two distinct vertices v and w in connected graph G is equal to the minimum number of edges whose removal disconnecting v and w.

14. 9.2 Menger’s theorem Theorem 9.6’ 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.

15. 9.2 Menger’s theorem Theorem 9.7’ A graph of order 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. Theorem 9.7 The maximum number of vertex-disjoint paths connecting two distinct non-adjacent vertices v and w of a connected graph G is equal to the minimum number of vertices whose removal disconnecting v and w.

16. 9.2 Menger’s theorem Theorem 9.8 The maximum number of arc-disjoint paths from a vertex v to a vertex w in a digraph D is equal to the minimum number of arcs whose removal disconnecting v and w.

17. 9.3 Menger’s Theorem implies the Max-flow min-cut Theorem Proof: Assuming the capacity of every arc is an integer. The network N can be seen as a digraph D in which • the capacities represent the number of arcs connecting the various vertices. • The maximum flow corresponds to the total number of arc-disjoint path form s to t in D; • The capacity of a minimum cut refers to the minimum number of arcs in a st-disconnecting set of D. N

18. 9.3 The Max-flow min-cut theorem implies Menger’s theorem Lemma 9.9 Let N be a network with source s and sink t in which each arc has unit capacity. Then • The value of a maximum flow in N is equal to the maximum number m of arc-disjoint directed (s,t)-paths in N; and • The capacity of a minimum cut in N is equal to the minimum number n of arcs whose deletion destroys all directed (s,t)-paths in N.

19. 9.3 Menger’s theorem Theorem 9.8 (Menger) Let s and t be two vertices of a digraph D. Then the maximum number of arc-disjoint directed (s,t)-paths in D is equal to the minimum number of arcs whose deletion destroys all directed (s,t)-paths in D.

20. 9.3 Menger’s theorem Theorem 9.10 (Menger, undirected version) Let s and t be two vertices of a graph G. Then the maximum number of edge-disjoint (s,t)-paths in G is equal to the minimum number of edges whose deletion destroys all (s,t)-paths in G. Proof: Apply the directed version of Menger’s theorem to the associated digraph D(G) of G (an edge becomes two directed edges). There is a one-one correspondence between paths in G and D(G). See Bondy and Murty.

21. 9.3 Menger’s theorem Theorem 9.11 Let s and t be two vertices of a directed graph D such that s is not joined to t. Then the maximum number of vertex-disjoint (s,t)-paths in G is equal to the minimum number of vertices whose deletion destroys all directed (s,t)-paths in D. Proof: by converting it to the arc-version of Menger’s theorem.

22. Construct a new digraph D’ from D by splitting each vertex vV-{s,t} such that v becomes an arc v’->v’’, arcs leading to v now leading to v’ and arcs leaving v now leaving from v’’; • To each edge-disjoint (s,t)-path in D’ there corresponds a vertex-disjoint directed (s,t)-path in D;and vice verse; and See Bondy and Murty.

23. g1 b1 g2 b2 g3 v w b3 g4 b4 g5 9.4 Menger’s theorem implies Hall’s Theorem Theorem 9.12 Menger’s theorem implies Hall’s theorem. There is a complete matching from V1 to V2 if and only if the number of vertex-disjoint paths from v to w is equal to |V1|=k.

24. 9.4 Menger’s theorem implies Hall’s Theorem Proof : Let G=(V1,V2) be a bipartite graph. We have to prove that if |A||N(A)| for each subset A of V1, then there exists a complete matching form V1 to V2. We add two extra vertices v and w (see the graph on previous page). Using Menger’s theorem of the vertex form, it is enough to prove that every vw-separating set (whose removal disconnect v and w) contains at least |V1|=k vertices. Let S be a vw-separating set, consists of AV1 and B  V2. Since AB is a vw-separating set, there can be no edges joining a vertex of V1-A to a vertex of V2-B, that is, N(V1-A)B. It follows that (Hall’s condition) |V1-A|<=|N(V1-A)|<=|B|. so |S|=|AB|=|A|+|B|>=|V1|=k, as required.

25. 9.4 Menger’s theorem implies Hall’s Theorem 证明: 设G=(V1,V2)是一个二分图. 证明如果对于V1的任意子集A, |A| <= |N(A)|, 则存在V1至V2的完全匹配. 在G上添加两个结点v,w, 使得v与V1的所有结点相邻, w与V2 的所有结点相邻. 显然, 存在V1至V2的完全匹配当且仅当v至w的结点不相交道路数目是|V1|=k. 因为V1构成分割v,w的一个集合，所以，根据Menger定理, 只需证明分割v,w的最少结点数是k. 设S是分割v,w的结点集, 且有V1的子集A和V2的子集B构成. 因为AB是一个vw分离集, 不存在V1-A与V2-B之间的边, 所以 N(V1-A)B, 再根据 Hall条件， |V1-A|<=|N(V1-A)|<=|B|. 所以, |S|=|AB|=|A|+|B|>=|V1|=k.

26. 9.5 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.

27. Summary • Vertex cut and edge cut; • Connectivity and edge-connectivity; • Menger Theorem; • Equivalence of Menger Theorem and the Max-flow Min-cut Theorem; • Menger Theorem implies Hall Theorem.