Introduction to Graph Theory

1. Introduction to Graph Theory Lecture 09: Distance and Connectivity

2. Connectivity • Plays an important role in reliability of computer networks • Removal of one or more vertices will break the graphs into several components • In this lecture, we’ll discuss several connectivity concept

3. Cut Vertices and Bridges • Cut vertex: A vertex of which the deletion disconnects the graph. • End vertices cannot be cut vertices • A deletion of such a vertex increases the number of components of G • Bridge (cut edge): Removal of such an edge increases the number of components. • Every edge of a tree is a bridge

4. Vertex Connectivity • Denoted is the minimum number of vertices whose deletion disconnects G or makes G trivial. • If G is disconnected then • A vertex cutset contains vertices whose removal disconnects the graph. • A graph G is called k-connected for some positive integer k if • G has a cut vertex if and only if • The same terminology applies to the edges too.

5. Edge Connectivity • Denoted is the minimum number of edges whose deletion disconnects G or makes G trivial. • An edge cutset contains edges whose removal disconnects the graph. • A connected graph has a bridge if and only if

6. Example 4.3 • Find a minimal vertex cutset of order 1 and 2, and minimal edge cutset of size 2 and 4. a d c e b h j f g i

7. Relation of and • Theorem: Given a connected graph G, we have • Proof: • To see , we can simply remove the edges of the vertex with minimum degree. • If S is the edge cutset consisting of k edges, then removal of k suitably chosen vertices removes the edges of S. Thus

8. Blocks • A block is a maximal connected subgraph of G with no cut vertices. • What are the blocks for the graph below?

9. Blocks • Theorem: The center of a connected graph G belongs to a single block of G. (We call such a block central block) • Proof: By contradiction • If G is connected without cut vertex, then the statement is true • Assume that G has one cut vertex v, and removal of v results in two components H and J • Suppose that , s.t. and then

10. (cont) • implies for some z • Assume . This implies that v is on every y-z geodesic. • Then • This implies , so contradiction • Therefore x and y must be in the same components of G-v. • Which block is the central block of our previous graph?

11. Menger’s Theorem • Menger showed that the connectivity of a graph is related to the number of disjoint paths joining two vertices. • Two paths connecting u and v are internally disjoint u-v paths if they have no vertices in common other than u and v. • Two paths are edge disjoint if they have no edges in common. • A set S of vertices or edges separate u and v if every path connecting u and v passes through S.

12. Menger’s Theorem • Menger’s Theorem: Let u and v be distinct nonadjacent vertices in G. Then the maximum number of internally disjoint paths connecting u and v equals the minimum number of vertices in a set that separate u and v. • (Proof omitted)

13. Example 4.5 • Finding the minimum order separating set and a maximum set of internally disjoint u-v path. a f b i g v u e c h d

14. More Theorem • Let u and v be distinct nonadjacent vertices in G. Then the maximum number of edge-disjoint paths connecting u and v equals the minimum number of edges in a set that separate u and v (called minimum cut). • Let’s try out this theorem on the previous graph.

15. Application: Network Reliability • You can picture a “network” as a weighted graph, where the weights are probabilities. • Network reliability concerned with how well a given network can withstand failure of individual components of a system. • There are several reliability models

16. Edge-Failure Model • Assumptions: • Vertices totally immune to failure • All edges fail independently with equal probability • Failure happen simultaneously • A common problem: • K-terminal reliability: to determine the probability that a subset K of terminal vertices remain connected to one another.

17. Example • Find the probability that vertices u and v remain in the same component. u x 0.2 0.2 0.2 0.2 0.2 w v

18. Vertex-Failure Model • Assumptions: • Edges are perfectly reliable • Vertices fail independently with same probability • A common problem: • The probability that the network remain connected.