Daniel BaldwinCOSC 494 – Graph Theory

4/9/2014

DefinitionsHistory

Examplse (graphs, sample problems, etc)Applications

State of the art, open problemsReferencesHOmework

Connectivity

• 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

• Edge-connectivity - = Minimum size of a disconnecting set

• k-edge connected if every disconnecting set has at least k edges

• Edge cut –

Definitions

ExamplesConsider a bipartition X, Y of

Since every separating set contains either X or Y which are themselves a separating set, [1]

Examples

Harary [1962]

Example of Edge Cut

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

Definitions

• 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

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

Applications

X,Y Cuts

Menger’s Theorem:

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

• 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.

Menger’s Theorem (Edge)

Elias-Feinstein-Shannon [1956] and Ford-Fulkerson [1956] proved that

• 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 1956Additionally proven independent in same year by L.R. Ford, Jr and D.R. Fulkerson.

Max-flow Min-cut

• Consider the graph

Max-flow Min-cut

Feasible flow of one

This is a maximal flow, but not a maximum flow.

• 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->3Increase flow 0->3

• 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

Max-flow Min-cut

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

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.

Homework1) Prove Menger’s Theorem for edge connectivity, i.e.

Homework

Homework

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).

ReferencesBoykov, 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.