Presentation on theme: "Introduction to Graph Theory"— Presentation transcript:
1 Introduction to Graph Theory Lecture 09: Distance and Connectivity
2 ConnectivityPlays an important role in reliability of computer networksRemoval of one or more vertices will break the graphs into several componentsIn 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 verticesA deletion of such a vertex increases the number of components of GBridge (cut edge): Removal of such an edge increases the number of components.Every edge of a tree is a bridge
4 Vertex ConnectivityDenoted is the minimum number of vertices whose deletion disconnects G or makes G trivial.If G is disconnected thenA vertex cutset contains vertices whose removal disconnects the graph.A graph G is called k-connected for some positive integer k ifG has a cut vertex if and only ifThe same terminology applies to the edges too.
5 Edge ConnectivityDenoted 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 4.3Find a minimal vertex cutset of order 1 and 2, and minimal edge cutset of size 2 and 4.adcebhjfgi
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 BlocksA block is a maximal connected subgraph of G with no cut vertices.What are the blocks for the graph below?
9 BlocksTheorem: The center of a connected graph G belongs to a single block of G. (We call such a block central block)Proof: By contradictionIf G is connected without cut vertex, then the statement is trueAssume that G has one cut vertex v, and removal of v results in two components H and JSuppose that , s.t and then
10 (cont) Which block is the central block of our previous graph? implies for some zAssume This implies that v is on every y-z geodesic.ThenThis implies , so contradictionTherefore 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 TheoremMenger 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 TheoremMenger’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 4.5Finding the minimum order separating set and a maximum set of internally disjoint u-v path.afbigvuechd
14 More TheoremLet 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: A common problem: Vertices totally immune to failureAll edges fail independently with equal probabilityFailure happen simultaneouslyA common problem:K-terminal reliability: to determine the probability that a subset K of terminal vertices remain connected to one another.
17 ExampleFind the probability that vertices u and v remain in the same component.ux0.20.20.20.20.2wv
18 Vertex-Failure Model Assumptions: A common problem: Edges are perfectly reliableVertices fail independently with same probabilityA common problem:The probability that the network remain connected.