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Connectivity. Daniel Baldwin COSC 494 – Graph Theory 4/9/2014 Definitions History Examplse (graphs, sample problems, etc ) Applications State of the art, open problems References HOmework. Definitions. Separating Set Connectivity k-connected – Connectivity is at least k

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  1. Connectivity Daniel Baldwin COSC 494 – Graph Theory 4/9/2014 Definitions History Examplse (graphs, sample problems, etc) Applications State of the art, open problems References HOmework

  2. Definitions • Separating Set • Connectivity • k-connected – Connectivity is at least k • Induced subgraph – subgraph obtained by deleting a set of vertices • Disconnecting set (of edges)

  3. Definitions • Edge-connectivity - = Minimum size of a disconnecting set • k-edge connected if every disconnecting set has at least k edges • Edge cut –

  4. Examples Consider a bipartition X, Y of Since every separating set contains either X or Y which are themselves a separating set, [1]

  5. Examples Harary [1962]

  6. Example of Edge Cut

  7. Definitions • Block – A maximal connected subgraph of G that has no cut-vertex.

  8. Applications • Network fault tolerance • The more disjoint paths, the better • Two paths from are internally disjoint if they have no common vertex. • When G has internally disjoint paths, deletion of any one vertex can not separate u from v (0 from 6).

  9. Applications • When can the streets in a road network all be made one-way without making any location unreachable from some other location?

  10. X,Y Cuts Menger’s Theorem:

  11. Menger’s Theorem (Vertex) Let S = {3, 4, 6, 7} be an x,y-cut denoted by with each pairwise internally disjoint path from/to x,y being red, green, blue or yellow.

  12. Applying to Edges • Line Graph – L(G) – the graph whose vertices are edges. Represents the adjacencies between the edges of G. 1) Take the pairwise product of each adjacent vertex {01, 12, 13, 23} 2) For each adjacency in the original graph, create a new adjacency in L(G) such that each member of G is connected to its representation in the pairwise product.

  13. Menger’s Theorem (Edge) Elias-Feinstein-Shannon [1956] and Ford-Fulkerson [1956] proved that

  14. Max-flow Min-cut • Applies to diagrams (directed graphs) • Definition: • Network is a digraph with a nonnegative capacity c(e) on each edge e. • Source vertex s • Sink vertex t • Flow assigns a function to each edge. • represents the total flow on edges leaving v • represents the total flow on edges entering v • Flow is “feasible” if it satisfies • Capacity constraints • Conservation constraints Proven by P. Elias, A. Feinstein, C.E. Shannon in 1956 Additionally proven independent in same year by L.R. Ford, Jr and D.R. Fulkerson.

  15. Max-flow Min-cut • Consider the graph Feasible flow of one This is a maximal flow, but not a maximum flow.

  16. Max-flow Min-cut • Goal: Achieve maximum flow on this graph • How: Create an f-augmenting path from the source to sink such that for every edge E(P) (Def. 4.3.4) Decrease flow 4->3 Increase flow 0->3

  17. Max-flow Min-cut • Def. 4.3.6. In a Network, a source/sink cut [S, T] consists of the edges from a source set S to a sink set T, where S and T partition the set of nodes, with . The capacity of the cut, cap(S, T), is the total capacities on the edges of [S, T] • 4.3.11 Theorem (Ford and Fulkerson [1956]) • Max-flow Min-cut Theorem: • In every network, the maximum value of a feasible flow equals the minimum capacity of the source/sink cut. • Max-flow: The maximum flow of a graph • Min-cut: a “cut” on the graph crossing the fewest number of edges separating the source-set and the sink-set. The edges S->T in this set should have a tail in S and a head in T. The capacity of the minimum cut is the sum of all the outbound edges in the cut.

  18. Max-flow Min-cut

  19. Max-flow Min-cut -Add a source and sink vertex -Add edges going from X to X’ -Set capacity of each edge to one -Compute the maximum flow

  20. Open Problems / Current Research • Jaeger-Swart Conjecture – every snark has edge connectivity of at most 6. Snark - Connected, bridgeless, cubic graph with chromatic index less than 4. Max-Flow Min-Cut Uses experimental algorithms for energy minimization in computer vision applications. Max-Flow Min-Cut algorithm for determining the optimal transmission path in a wireless communication network.

  21. Homework 1) Prove Menger’s Theorem for edge connectivity, i.e.

  22. Homework

  23. Homework

  24. References [1] West, Douglas B. Introduction to Graph Theory, Second Edition. University of Illinois. 2001. Harary, F. The maximum connectivity of a graph. 1962. 1142-1146. Menger, Karl. ZurallgemeinenKurventheorie(On the general theory of curves). 1927. Schrijver, Alexander. Paths and Flows – A Historical Survey. University of Amsterdam. Ford and Fulkerson [1956] Eugene Lawler. Combinatorial Optimization: Networks and Matroids. (2001).

  25. References Boykov, Y. University of Western Ontario. An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. 2004. S. M. SadeghTabatabaeiYazdi and Serap A. Savari. 2010. A max-flow/min-cut algorithm for a class of wireless networks. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms (SODA '10). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1209-1226.

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