COSC 494 Graph Theory 492014 Definitions History Examplse graphs sample problems etc Applications State of the art open problems References HOmework Connectivity Separating Set ID: 468448
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Daniel BaldwinCOSC 494 – Graph Theory4/9/2014DefinitionsHistoryExamplse (graphs, sample problems, etc)ApplicationsState of the art, open problemsReferencesHOmework
ConnectivitySlide2
Separating SetConnectivityk-connected – Connectivity is at least kInduced subgraph – subgraph obtained by deleting a set of verticesDisconnecting set (of edges)
DefinitionsSlide3
Edge-connectivity - = Minimum size of a disconnecting setk-edge connected if every disconnecting set has at least k edges Edge cut –
DefinitionsSlide4
Examples
Consider a bipartition X, Y of
Since every separating set contains either X or Y which are themselves a separating set,
[1]Slide5
ExamplesHarary [1962]Slide6
Example of Edge CutSlide7
Block – A maximal connected subgraph of G that has no cut-vertex.DefinitionsSlide8
Network fault toleranceThe more disjoint paths, the betterTwo 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).
ApplicationsSlide9
When can the streets in a road network all be made one-way without making any location unreachable from some other location? ApplicationsSlide10
X,Y Cuts
Menger’s Theorem:Slide11
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.Slide12
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. Slide13
Menger’s Theorem (Edge)
Elias-Feinstein-Shannon [1956] and Ford-Fulkerson [1956] proved that
Slide14
Applies to diagrams (directed graphs)Definition: Network is a digraph with a nonnegative capacity c(e) on each edge e.Source vertex sSink vertex tFlow assigns a function to each edge. represents the total flow on edges leaving v represents the total flow on edges entering vFlow is “feasible” if it satisfies
C
apacity 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-cutSlide15
Consider the graphMax-flow Min-cutFeasible flow of one
This is a maximal flow, but not a maximum flow.Slide16
Goal: Achieve maximum flow on this graphHow: 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->3Slide17
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-cutSlide18
Max-flow Min-cutSlide19
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 flowSlide20
Open Problems / Current ResearchJaeger-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. Slide21
Homework 1) Prove Menger’s Theorem for edge connectivity, i.e. Slide22
Homework Slide23
HomeworkSlide24
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). Slide25
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.