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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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connectivity
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
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)
definitions1
Definitions
  • Edge-connectivity - = Minimum size of a disconnecting set
  • k-edge connected if every disconnecting set has at least k edges
  • Edge cut –
examples
Examples

Consider a bipartition X, Y of

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

examples1
Examples

Harary [1962]

definitions2
Definitions
  • Block – A maximal connected subgraph of G that has no cut-vertex.
applications
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).
applications1
Applications
  • When can the streets in a road network all be made one-way without making any location unreachable from some other location?
x y cuts
X,Y Cuts

Menger’s Theorem:

menger s theorem vertex
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.

applying to edges
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.

menger s theorem edge
Menger’s Theorem (Edge)

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

max flow min cut
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.

max flow min cut1
Max-flow Min-cut
  • Consider the graph

Feasible flow of one

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

max flow min cut2
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

max flow min cut3
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.
max flow min cut5
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
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.

homework
Homework

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

references
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).

references1
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.