4.2 k-connected graphs This copyrighted material is taken from Introduction to Graph Theory, 2 nd Ed., by Doug West; and is not for further distribution.

Slides:

Advertisements

Similar presentations

Discrete Mathematics University of Jazeera College of Information Technology & Design Khulood Ghazal Connectivity Lecture _13.

Advertisements

Connectivity - Menger’s Theorem Graphs & Algorithms Lecture 3.

Chapter 8 Topics in Graph Theory

Edge-connectivity and super edge-connectivity of P 2 -path graphs Camino Balbuena, Daniela Ferrero Discrete Mathematics 269 (2003) 13 – 20.

22C:19 Discrete Math Graphs Fall 2010 Sukumar Ghosh.

22C:19 Discrete Math Graphs Fall 2014 Sukumar Ghosh.

Bipartite Matching, Extremal Problems, Matrix Tree Theorem.

Walks, Paths and Circuits Walks, Paths and Circuits Sanjay Jain, Lecturer, School of Computing.

Introduction to Graphs

Convex drawing chapter 5 Ingeborg Groeneweg. Summery What is convex drawing What is convex drawing Some definitions Some definitions Testing convexity.

EMIS 8374 Vertex Connectivity Updated 20 March 2008.

Chapter 9 Connectivity 连通度. 9.1 Connectivity Consider the following graphs:  G 1 : Deleting any edge makes it disconnected.  G 2 : Cannot be disconnected.

Graph Theory Ming-Jer Tsai. Outline Text Book Graph Graph Theory - Course Description The Topics in the Class Evaluation.

Vertex Cut Vertex Cut: A separating set or vertex cut of a graph G is a set S  V(G) such that G-S has more than one component. a b c d e f g h i.

Internally Disjoint Paths Internally Disjoint Paths : Two paths u to v are internally disjoint if they have no common internal vertex.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

Graph Theory Ming-Jer Tsai. Outline Graph Graph Theory Grades Q & A.

Graph Orientations and Submodular Flows Lecture 6: Jan 26.

Vertex Cut Vertex Cut: A separating set or vertex cut of a graph G is a set SV(G) such that S has more than one component. Connectivity of G ((G)): The.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

Definition Dual Graph G* of a Plane Graph:

Internally Disjoint Paths

Internally Disjoint Paths

A 2-Approximation algorithm for finding an optimum 3-Vertex-Connected Spanning Subgraph.

1 Submodular Functions in Combintorial Optimization Lecture 6: Jan 26 Lecture 8: Feb 1.

1 Section 8.4 Connectivity. 2 Paths In an undirected graph, a path of length n from u to v, where n is a positive integer, is a sequence of edges e 1,

What is the next line of the proof? a). Assume the theorem holds for all graphs with k edges. b). Let G be a graph with k edges. c). Assume the theorem.

Internally Disjoint Paths Internally Disjoint Paths : Two paths u to v are internally disjoint if they have no common internal vertex. u u v v Common internal.

Vertex Cut Vertex Cut: A separating set or vertex cut of a graph G is a set SV(G) such that G-S has more than one component. d f b e a g c i h.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 10, Monday, September 22.

22C:19 Discrete Math Graphs Spring 2014 Sukumar Ghosh.

Daniel Baldwin COSC 494 – Graph Theory 4/9/2014 Definitions History Examplse (graphs, sample problems, etc) Applications State of the art, open problems.

Introduction to Graph Theory

GRAPH Learning Outcomes Students should be able to:

Subdivision of Edge In a graph G, subdivision of an edge uv is the operation of replacing uv with a path u,w,v through a new vertex w.

3.3 Matchings and Factors: Matchings in General Graphs This copyrighted material is taken from Introduction to Graph Theory, 2 nd Ed., by Doug West; and.

Chapter 1 Fundamental Concepts II Pao-Lien Lai 1.

4.1 Connectivity and Paths: Cuts and Connectivity

4.2 k-connected graphs This copyrighted material is taken from Introduction to Graph Theory, 2 nd Ed., by Doug West; and is not for further distribution.

Data Structures & Algorithms Graphs

The countable character of uncountable graphs François Laviolette Barbados 2003.

3.2 Matchings and Factors: Algorithms and Applications This copyrighted material is taken from Introduction to Graph Theory, 2 nd Ed., by Doug West; and.

Chapter 1 Fundamental Concepts Introduction to Graph Theory Douglas B. West July 11, 2002.

10. Lecture WS 2014/15 Bioinformatics III1 V10 Metabolic networks - Graph connectivity Graph connectivity is related to analyzing biological networks for.

Connectivity and Paths 報告人：林清池. Connectivity A separating set of a graph G is a set such that G-S has more than one component. The connectivity of G,

Chapter 6 Connectivity and Flow 大葉大學資訊工程系黃鈴玲

CSCI 115 Chapter 8 Topics in Graph Theory. CSCI 115 §8.1 Graphs.

4.2 k-connected graphs This copyrighted material is taken from Introduction to Graph Theory, 2 nd Ed., by Doug West; and is not for further distribution.

Graphs Upon completion you will be able to:

Introduction to Graph Theory

Graph Theory Ming-Jer Tsai. Outline Graph Graph Theory Grades Q & A.

8.5 Equivalence Relations

12. Lecture WS 2012/13Bioinformatics III1 V12 Menger’s theorem Borrowing terminology from operations research consider certain primal-dual pairs of optimization.

Main Index Contents 11 Main Index Contents Graph Categories Graph Categories Example of Digraph Example of Digraph Connectedness of Digraph Connectedness.

1 GRAPH Learning Outcomes Students should be able to: Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using.

V15: Max-Flow Min-Cut V15 continues chapter 12 in Gross & Yellen „Graph Theory“ Theorem [Characterization of Maximum Flow] Let f be a flow in a.

Graph Graphs and graph theory can be used to model:

The countable character of uncountable graphs François Laviolette Barbados 2003.

What is the next line of the proof?

Planarity Testing.

V17 Metabolic networks - Graph connectivity

V11 Metabolic networks - Graph connectivity

Mathematics for Computer Science MIT 6.042J/18.062J

V12 Menger’s theorem Borrowing terminology from operations research

Graphs G = (V, E) V are the vertices; E are the edges.

V11 Metabolic networks - Graph connectivity

Simple Graphs: Connectedness, Trees

Graph Theory: Cuts and Connectivity

V11 Metabolic networks - Graph connectivity

Treewidth meets Planarity

Presentation transcript:

4.2 k-connected graphs This copyrighted material is taken from Introduction to Graph Theory, 2 nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 1

4.2.9 Definition. A closed ear in a graph G is a cycle C such that all vertices of C except one have degree 2 in G. Example A decomposition using closed ears and (open) ears Closed Ears versus (Open) Ears 2 P0P0 P 1 (open) P 2 (closed) P 3 (open) Contains copyrighted material from Introduction to Graph Theory by Doug West, 2 nd Ed. Not for distribution beyond IIT’s Math 454/553. P 4 (closed)

A closed-ear decomposition of a graph G is a decomposition P 0,P 1,…,P i,…,P k such that P 0 is a cycle and P i for i > 0 is either an (open) ear or a closed ear in the union of P 0,P 1,…,P i Theorem. A graph is 2-connected iff it has a closed-ear decomposition, and every cycle in a 2-edge-connected graph is the initial cycle in some such decomposition. The proof of Theorem is quite similar to that of Theorem (with 2-connected iff ear decomposition). See p164. (=>) Show 2-edge-connectectedness is maintained on addition of each P i. (<=) Find a cycle P 0 in G. Iteratively find and add ears P i containing an edge with one endpoint in the graph so far. Closed-Ear Decompositions and 2-Edge-Connectivity 3 Contains copyrighted material from Introduction to Graph Theory by Doug West, 2 nd Ed. Not for distribution beyond IIT’s Math 454/553.

Definition. For a digraph D: In digraphs, removal of separating sets (vertex cuts) destroys strong connectivity, or reduces the graph to 1 vertex. D – S could still be weakly connected for a vertex cut S! Connectivity κ(D) and k-connectedness are defined as for graphs. Edge cuts [S,V(D)–S] are all edges with tail in S and head in V(D)–S, for proper, nonempty subsets S of V(D). Edge connectivity κ’(D) and k-edge-connectedness are defined as for graphs Theorem. (Robbins 1939) A graph has a strong orientation iff it is 2-edge-connected. Connectivity of Digraphs 4 Contains copyrighted material from Introduction to Graph Theory by Doug West, 2 nd Ed. Not for distribution beyond IIT’s Math 454/553. S V(D)–S

Local connectivity considers the number of alternative paths between a pair of vertices x,y, and the minimum size structure needed to be deleted to disconnect x from y. Global connectivity considers the minimum number of alternative paths between any pair of vertices, and the minimum size structure needed to be deleted to disconnect the graph. We have seen that (for n(G) ¸ 3) 2-connectedness is equivalent to there existing 2 internally disjoint paths between any pair of vertices. We want to extend this idea: Locally: compare alternative x,y-paths versus minimum x,y-cuts Globally: find vertex pairs optimizing the local values Local connectivity and Menger’s Theorem 5 Contains copyrighted material from Introduction to Graph Theory by Doug West, 2 nd Ed. Not for distribution beyond IIT’s Math 454/553.