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
•
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 –
Examples
Consider 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
Definitions
•
Block – A maximal connected subgraph of G that has no cut-vertex.
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).
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
Menger’s Theorem:
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
•
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)
Elias-Feinstein-Shannon [1956] and Ford-Fulkerson [1956] proved that
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-cut
•
Consider the graph
Feasible flow of one
This is a maximal flow,
but not a maximum
flow.
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-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-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.
Homework
1) 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).
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 maxflow/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.