1. 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 Connectivity

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

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

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

8. 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). Applications

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

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. Line Graph – L(G) – the graph whose vertices are edges. Represents the adjacencies between the edges of G. Applying to Edges 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. 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. Max-flow Min-cut

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

16. 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) – Max-flow Min-cut Decrease flow 4->3 Increase flow 0->3

17. 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. Max-flow Min-cut

19. -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

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. Zur allgemeinen Kurventheorie (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. Sadegh Tabatabaei Yazdi 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.