Paths. Path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.
|
|
- Claud King
- 5 years ago
- Views:
Transcription
1 Paths Path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.
2 Formal Definition of a Path (Undirected) Let n be a nonnegative integer and G an undirected graph. A path of length n from u to v in G is a sequence of n edges e 1,, e n of G for which there exists a sequence x 0 = u, x 1, x n-1, x n = v of vertices such that e i has, for i = 1,, n, the endpoints x i-1 and x i. When the graph is simple, we denote this path by its vertex sequence x 0, x 1,, x n (because listing these vertices uniquely determines the path) The path is a circuit if begins and ends at the same vertex (if u = v and length > 0) The path is said to pass through the vertices x 1, x 2,, x n-1 or traverse edges e 1, e 2, e n A path or circuit is simple if it does not contain the same edge more than once.
3 Formal Definition of a Path (Directed) Let n be a nonnegative integer and G a directed graph. A path of length n from u to v in G is a sequence of edges e 1, e 2,, e n of G such that e 1 is associated with (x 0, x 1 ), e 2 is associated with (x 1, x 2 ) and so on, with e n associated with (x n-1, x n ), where x 0 = u and x n = v. When there are no multiple edges in the directed graph, this path is denoted by its vertex sequence x 0, x 1, x 2,, x n. A path of length greater than zero that begins and ends at the same vertex is called a circuit or cycle. A path or circuit is called simple if it does not contain the same edge more than once.
4 Connectedness in Undirected Graphs An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. An undirected graph that is not connected is called disconnected. We say that we disconnect a graph when we remove vertices or edges or both to produce a disconnected subgraph.
5 Theorem There is a simple path between every pair of distinct vertices of a connected undirected graph.
6 Connected Components A connected component of a graph G is a connected subgraph of G that is not a proper subgraph of another connected subgraph of G. That is, a connected component of a graph G is a maximal connected subgraph of G A graph G that is not connected has two or more connected components that are disjoint and have G as their union.
7 Cut Vertices and Edges A vertex whose removal, along with the removal of all incident edges, produces a subgraph with more connected components is called a cut vertex. Connected graphs without cut vertices are called non-separable graphs. An edge whose removal produces a graph with more connected components than the original graph is called a cut edge
8 Vertex Connectivity A subset V of the vertex set V of G = (V, E) is a vertex cut if G V is disconnected We define the vertex connectivity of a noncomplete graph G, denoted by κ(g), as the minimum number of vertices in a vertex cut. We say that a graph is k-connected if κ(g) k
9 Edge Connectivity A set of edges E is called an edge cut of G if the subgraph G E is disconnected. The edge connectivity of a graph, denoted by λ(g), is the minimum number of edges in an edge cut of G. Inequality for Vertex/Edge Connectivity κ(g) λ(g) min(deg(v)) for v ϵ V
10 Connectivity in Directed Graphs A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph A directed graph is weakly connected if there is a path between every two vertices in the underlying undirected graph
11 Strongly Connected Components A subgraph that is strongly connected, but not contained in a larger strongly connected subgraph is called a strongly connected component.
12 Paths and Isomorphism The existence of a simple circuit of a particular length is an invariant that can be used to show that two graphs are not isomorphic Paths can also be used as a guideline when constructing an isomorphism between two graphs.
13 Counting Paths between Vertices Theorem: Let G be a graph with adjacency matrix A with respect to the ordering v 1, v 2,, v n of vertices of the graph (with directed or undirected edges, with multiple edges and loops allowed). The number of different paths of length r from v i to v j, where r is a positive integer equals the (i, j) th entry of A r.
CS 311 Discrete Math for Computer Science Dr. William C. Bulko. Graphs
CS 311 Discrete Math for Computer Science Dr. William C. Bulko Graphs 2014 Definitions Definition: A graph G = (V,E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge
More informationGraphs. Introduction To Graphs: Exercises. Definitions:
Graphs Eng.Jehad Aldahdooh Introduction To Graphs: Definitions: A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated
More information2. CONNECTIVITY Connectivity
2. CONNECTIVITY 70 2. Connectivity 2.1. Connectivity. Definition 2.1.1. (1) A path in a graph G = (V, E) is a sequence of vertices v 0, v 1, v 2,..., v n such that {v i 1, v i } is an edge of G for i =
More informationGraphs. Pseudograph: multiple edges and loops allowed
Graphs G = (V, E) V - set of vertices, E - set of edges Undirected graphs Simple graph: V - nonempty set of vertices, E - set of unordered pairs of distinct vertices (no multiple edges or loops) Multigraph:
More informationIntroduction to Graphs
Graphs Introduction to Graphs Graph Terminology Directed Graphs Special Graphs Graph Coloring Representing Graphs Connected Graphs Connected Component Reading (Epp s textbook) 10.1-10.3 1 Introduction
More informationPaths, Circuits, and Connected Graphs
Paths, Circuits, and Connected Graphs Paths and Circuits Definition: Let G = (V, E) be an undirected graph, vertices u, v V A path of length n from u to v is a sequence of edges e i = {u i 1, u i} E for
More informationDiscrete Structures CISC 2315 FALL Graphs & Trees
Discrete Structures CISC 2315 FALL 2010 Graphs & Trees Graphs A graph is a discrete structure, with discrete components Components of a Graph edge vertex (node) Vertices A graph G = (V, E), where V is
More informationVarying Applications (examples)
Graph Theory Varying Applications (examples) Computer networks Distinguish between two chemical compounds with the same molecular formula but different structures Solve shortest path problems between cities
More informationNumber Theory and Graph Theory
1 Number Theory and Graph Theory Chapter 6 Basic concepts and definitions of graph theory By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com
More informationGRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS
GRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS DR. ANDREW SCHWARTZ, PH.D. 10.1 Graphs and Graph Models (1) A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes)
More informationGraph Theory S 1 I 2 I 1 S 2 I 1 I 2
Graph Theory S I I S S I I S Graphs Definition A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) ( ) V (G). x and y are the endpoints of edge e = {x, y}. They are called adjacent
More informationCSC Intro to Intelligent Robotics, Spring Graphs
CSC 445 - Intro to Intelligent Robotics, Spring 2018 Graphs Graphs Definition: A graph G = (V, E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge has either one or two
More informationIntroduction III. Graphs. Motivations I. Introduction IV
Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg
More information1 Digraphs. Definition 1
1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together
More informationGraph and Digraph Glossary
1 of 15 31.1.2004 14:45 Graph and Digraph Glossary A B C D E F G H I-J K L M N O P-Q R S T U V W-Z Acyclic Graph A graph is acyclic if it contains no cycles. Adjacency Matrix A 0-1 square matrix whose
More informationIntroduction to Graph Theory
Introduction to Graph Theory Tandy Warnow January 20, 2017 Graphs Tandy Warnow Graphs A graph G = (V, E) is an object that contains a vertex set V and an edge set E. We also write V (G) to denote the vertex
More informationCMSC 380. Graph Terminology and Representation
CMSC 380 Graph Terminology and Representation GRAPH BASICS 2 Basic Graph Definitions n A graph G = (V,E) consists of a finite set of vertices, V, and a finite set of edges, E. n Each edge is a pair (v,w)
More information(5.2) 151 Math Exercises. Graph Terminology and Special Types of Graphs. Malek Zein AL-Abidin
King Saud University College of Science Department of Mathematics 151 Math Exercises (5.2) Graph Terminology and Special Types of Graphs Malek Zein AL-Abidin ه Basic Terminology First, we give some terminology
More informationCS 441 Discrete Mathematics for CS Lecture 26. Graphs. CS 441 Discrete mathematics for CS. Final exam
CS 441 Discrete Mathematics for CS Lecture 26 Graphs Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Final exam Saturday, April 26, 2014 at 10:00-11:50am The same classroom as lectures The exam
More informationLet G = (V, E) be a graph. If u, v V, then u is adjacent to v if {u, v} E. We also use the notation u v to denote that u is adjacent to v.
Graph Adjacent Endpoint of an edge Incident Neighbors of a vertex Degree of a vertex Theorem Graph relation Order of a graph Size of a graph Maximum and minimum degree Let G = (V, E) be a graph. If u,
More informationCharacterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)
S-72.2420/T-79.5203 Basic Concepts 1 S-72.2420/T-79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G
More informationPart II. Graph Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 53 Paper 3, Section II 15H Define the Ramsey numbers R(s, t) for integers s, t 2. Show that R(s, t) exists for all s,
More informationUNDIRECTED GRAPH: a set of vertices and a set of undirected edges each of which is associated with a set of one or two of these vertices.
Graphs 1 Graph: A graph G = (V, E) consists of a nonempty set of vertices (or nodes) V and a set of edges E. Each edge has either one or two vertices associated with it, called its endpoints. An edge is
More informationGRAPH THEORY - CONNECTIVITY
GRAPH THEORY - CONNECTIVITY http://www.tutorialspoint.com/graph_theory/graph_theory_connectivity.htm Copyright tutorialspoint.com Whether it is possible to traverse a graph from one vertex to another is
More informationAssignment 4 Solutions of graph problems
Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of
More informationFundamental Properties of Graphs
Chapter three In many real-life situations we need to know how robust a graph that represents a certain network is, how edges or vertices can be removed without completely destroying the overall connectivity,
More informationAlgorithms. Graphs. Algorithms
Algorithms Graphs Algorithms Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Algorithms 1 Graphs Vertices: Nodes, points, computers, users, items,...
More informationGraphs: Introduction. Ali Shokoufandeh, Department of Computer Science, Drexel University
Graphs: Introduction Ali Shokoufandeh, Department of Computer Science, Drexel University Overview of this talk Introduction: Notations and Definitions Graphs and Modeling Algorithmic Graph Theory and Combinatorial
More informationBasics of Graph Theory
Basics of Graph Theory 1 Basic notions A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Simple graphs have their
More informationCONNECTIVITY AND NETWORKS
CONNECTIVITY AND NETWORKS We begin with the definition of a few symbols, two of which can cause great confusion, especially when hand-written. Consider a graph G. (G) the degree of the vertex with smallest
More informationAlgorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95
Algorithms: Graphs Amotz Bar-Noy CUNY Spring 2012 Amotz Bar-Noy (CUNY) Graphs Spring 2012 1 / 95 Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Amotz
More informationAdjacent: Two distinct vertices u, v are adjacent if there is an edge with ends u, v. In this case we let uv denote such an edge.
1 Graph Basics What is a graph? Graph: a graph G consists of a set of vertices, denoted V (G), a set of edges, denoted E(G), and a relation called incidence so that each edge is incident with either one
More information4. (a) Draw the Petersen graph. (b) Use Kuratowski s teorem to prove that the Petersen graph is non-planar.
UPPSALA UNIVERSITET Matematiska institutionen Anders Johansson Graph Theory Frist, KandMa, IT 010 10 1 Problem sheet 4 Exam questions Solve a subset of, say, four questions to the problem session on friday.
More informationCPCS Discrete Structures 1
Let us switch to a new topic: Graphs CPCS 222 - Discrete Structures 1 Introduction to Graphs Definition: A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs
More informationGraph Theory CS/Math231 Discrete Mathematics Spring2015
1 Graphs Definition 1 A directed graph (or digraph) G is a pair (V, E), where V is a finite set and E is a binary relation on V. The set V is called the vertex set of G, and its elements are called vertices
More informationDomination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes
Domination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes Leonor Aquino-Ruivivar Mathematics Department, De La Salle University Leonorruivivar@dlsueduph
More informationCompatible circuits in eulerian digraphs
Compatible circuits in eulerian digraphs James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Stephen Hartke March 2012 James Carraher (UNL) Compatible circuits in eulerian
More informationAssignment 1 Introduction to Graph Theory CO342
Assignment 1 Introduction to Graph Theory CO342 This assignment will be marked out of a total of thirty points, and is due on Thursday 18th May at 10am in class. Throughout the assignment, the graphs are
More informationElements of Graph Theory
Elements of Graph Theory Quick review of Chapters 9.1 9.5, 9.7 (studied in Mt1348/2008) = all basic concepts must be known New topics we will mostly skip shortest paths (Chapter 9.6), as that was covered
More informationK 4 C 5. Figure 4.5: Some well known family of graphs
08 CHAPTER. TOPICS IN CLASSICAL GRAPH THEORY K, K K K, K K, K K, K C C C C 6 6 P P P P P. Graph Operations Figure.: Some well known family of graphs A graph Y = (V,E ) is said to be a subgraph of a graph
More informationDiscrete Mathematics (2009 Spring) Graphs (Chapter 9, 5 hours)
Discrete Mathematics (2009 Spring) Graphs (Chapter 9, 5 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University June 1, 2009 9.1 Graphs and Graph Models What are Graphs? General meaning
More informationModule 11. Directed Graphs. Contents
Module 11 Directed Graphs Contents 11.1 Basic concepts......................... 256 Underlying graph of a digraph................ 257 Out-degrees and in-degrees.................. 258 Isomorphism..........................
More informationGraph Theory. Connectivity, Coloring, Matching. Arjun Suresh 1. 1 GATE Overflow
Graph Theory Connectivity, Coloring, Matching Arjun Suresh 1 1 GATE Overflow GO Classroom, August 2018 Thanks to Subarna/Sukanya Das for wonderful figures Arjun, Suresh (GO) Graph Theory GATE 2019 1 /
More informationAn Introduction to Graph Theory
An Introduction to Graph Theory CIS008-2 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 12:00, Friday 17 th February 2012 Outline 1 Graphs 2 Paths and cycles 3 Graphs and
More informationWUCT121. Discrete Mathematics. Graphs
WUCT121 Discrete Mathematics Graphs WUCT121 Graphs 1 Section 1. Graphs 1.1. Introduction Graphs are used in many fields that require analysis of routes between locations. These areas include communications,
More information22 Elementary Graph Algorithms. There are two standard ways to represent a
VI Graph Algorithms Elementary Graph Algorithms Minimum Spanning Trees Single-Source Shortest Paths All-Pairs Shortest Paths 22 Elementary Graph Algorithms There are two standard ways to represent a graph
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. An Introduction to Graph Theory
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics An Introduction to Graph Theory. Introduction. Definitions.. Vertices and Edges... The Handshaking Lemma.. Connected Graphs... Cut-Points and Bridges.
More informationV :non-empty vertex ornode set E V V :edge set G (V, E) :directed graph on V, or digraph on V
-93-11. Graph Theory Example: V :non-empty vertex ornode set E V V :edge set G (V, E) :directed graph on V, or digraph on V b e f V={a, b, c, d, e, f, g} a c d f E={(a,b), (b,c), (c,a),... } Note: (a,
More informationOutline. Introduction. Representations of Graphs Graph Traversals. Applications. Definitions and Basic Terminologies
Graph Chapter 9 Outline Introduction Definitions and Basic Terminologies Representations of Graphs Graph Traversals Breadth first traversal Depth first traversal Applications Single source shortest path
More informationDEFINITION OF GRAPH GRAPH THEORY GRAPHS ACCORDING TO THEIR VERTICES AND EDGES EXAMPLE GRAPHS ACCORDING TO THEIR VERTICES AND EDGES
DEFINITION OF GRAPH GRAPH THEORY Prepared by Engr. JP Timola Reference: Discrete Math by Kenneth H. Rosen A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each
More informationWeek 9-10: Connectivity
Week 9-0: Connectiity October 3, 206 Vertex Connectiity Let G = (V, E) be a graph. Gien two ertices x, y V. Two (x, y)-path are said to be internally disjoint if they hae no internal ertices in common.
More informationCMSC 451: Lecture 22 Approximation Algorithms: Vertex Cover and TSP Tuesday, Dec 5, 2017
CMSC 451: Lecture 22 Approximation Algorithms: Vertex Cover and TSP Tuesday, Dec 5, 2017 Reading: Section 9.2 of DPV. Section 11.3 of KT presents a different approximation algorithm for Vertex Cover. Coping
More information11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions
Introduction Chapter 9 Graph Algorithms graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 2 Definitions an undirected graph G = (V, E) is
More informationChapter 9 Graph Algorithms
Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 3 Definitions an undirected graph G = (V, E)
More informationBest known solution time is Ω(V!) Check every permutation of vertices to see if there is a graph edge between adjacent vertices
Hard Problems Euler-Tour Problem Undirected graph G=(V,E) An Euler Tour is a path where every edge appears exactly once. The Euler-Tour Problem: does graph G have an Euler Path? Answerable in O(E) time.
More information22 Elementary Graph Algorithms. There are two standard ways to represent a
VI Graph Algorithms Elementary Graph Algorithms Minimum Spanning Trees Single-Source Shortest Paths All-Pairs Shortest Paths 22 Elementary Graph Algorithms There are two standard ways to represent a graph
More informationGraph theory - solutions to problem set 1
Graph theory - solutions to problem set 1 1. (a) Is C n a subgraph of K n? Exercises (b) For what values of n and m is K n,n a subgraph of K m? (c) For what n is C n a subgraph of K n,n? (a) Yes! (you
More informationCS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS
CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1. Define Graph. A graph G = (V, E) consists
More informationLecture 4: Walks, Trails, Paths and Connectivity
Lecture 4: Walks, Trails, Paths and Connectivity Rosa Orellana Math 38 April 6, 2015 Graph Decompositions Def: A decomposition of a graph is a list of subgraphs such that each edge appears in exactly one
More informationGraph Theory: Introduction
Graph Theory: Introduction Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in Resources Copies of slides available at: http://www.facweb.iitkgp.ernet.in/~pallab
More informationMatching and Planarity
Matching and Planarity Po-Shen Loh June 010 1 Warm-up 1. (Bondy 1.5.9.) There are n points in the plane such that every pair of points has distance 1. Show that there are at most n (unordered) pairs of
More informationChapter 2 Graphs. 2.1 Definition of Graphs
Chapter 2 Graphs Abstract Graphs are discrete structures that consist of vertices and edges connecting some of these vertices. Graphs have many applications in Mathematics, Computer Science, Engineering,
More informationCS473-Algorithms I. Lecture 13-A. Graphs. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorithms I Lecture 3-A Graphs Graphs A directed graph (or digraph) G is a pair (V, E), where V is a finite set, and E is a binary relation on V The set V: Vertex set of G The set E: Edge set of
More informationHW Graph Theory SOLUTIONS (hbovik)
Diestel 1.3: Let G be a graph containing a cycle C, and assume that G contains a path P of length at least k between two vertices of C. Show that G contains a cycle of length at least k. If C has length
More informationInf 496/596 Topics in Informatics: Analysis of Social Network Data
Inf 496/596 Topics in Informatics: Analysis of Social Network Data Jagdish S. Gangolly Department of Informatics College of Computing & Information State University of New York at Albany Lecture 1B (Graphs)
More informationDHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI
DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI Department of Computer Science and Engineering CS6702 - GRAPH THEORY AND APPLICATIONS Anna University 2 & 16 Mark Questions & Answers Year / Semester: IV /
More informationEulerian circuits with no monochromatic transitions
Eulerian circuits with no monochromatic transitions James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Stephen Hartke June 2012 James Carraher (UNL) Eulerian circuits
More informationDS UNIT 4. Matoshri College of Engineering and Research Center Nasik Department of Computer Engineering Discrete Structutre UNIT - IV
Sr.No. Question Option A Option B Option C Option D 1 2 3 4 5 6 Class : S.E.Comp Which one of the following is the example of non linear data structure Let A be an adjacency matrix of a graph G. The ij
More informationMath 170- Graph Theory Notes
1 Math 170- Graph Theory Notes Michael Levet December 3, 2018 Notation: Let n be a positive integer. Denote [n] to be the set {1, 2,..., n}. So for example, [3] = {1, 2, 3}. To quote Bud Brown, Graph theory
More informationGraphs and Trees. An example. Graphs. Example 2
Graphs and Trees An example How would you describe this network? What kind of model would you write for it? What kind of information would you expect to obtain? Relationship between some of the apoptotic
More informationγ(ɛ) (a, b) (a, d) (d, a) (a, b) (c, d) (d, d) (e, e) (e, a) (e, e) (a) Draw a picture of G.
MAD 3105 Spring 2006 Solutions for Review for Test 2 1. Define a graph G with V (G) = {a, b, c, d, e}, E(G) = {r, s, t, u, v, w, x, y, z} and γ, the function defining the edges, is given by the table ɛ
More informationChapter 3: Paths and Cycles
Chapter 3: Paths and Cycles 5 Connectivity 1. Definitions: Walk: finite sequence of edges in which any two consecutive edges are adjacent or identical. (Initial vertex, Final vertex, length) Trail: walk
More informationV10 Metabolic networks - Graph connectivity
V10 Metabolic networks - Graph connectivity Graph connectivity is related to analyzing biological networks for - finding cliques - edge betweenness - modular decomposition that have been or will be covered
More informationMinimum-Spanning-Tree problem. Minimum Spanning Trees (Forests) Minimum-Spanning-Tree problem
Minimum Spanning Trees (Forests) Given an undirected graph G=(V,E) with each edge e having a weight w(e) : Find a subgraph T of G of minimum total weight s.t. every pair of vertices connected in G are
More informationTheorem 3.1 (Berge) A matching M in G is maximum if and only if there is no M- augmenting path.
3 Matchings Hall s Theorem Matching: A matching in G is a subset M E(G) so that no edge in M is a loop, and no two edges in M are incident with a common vertex. A matching M is maximal if there is no matching
More informationMTL 776: Graph Algorithms. B S Panda MZ 194
MTL 776: Graph Algorithms B S Panda MZ 194 bspanda@maths.iitd.ac.in bspanda1@gmail.com Lectre-1: Plan Definitions Types Terminology Sb-graphs Special types of Graphs Representations Graph Isomorphism Definitions
More informationMa/CS 6b Class 5: Graph Connectivity
Ma/CS 6b Class 5: Graph Connectivity By Adam Sheffer Communications Network We are given a set of routers and wish to connect pairs of them to obtain a connected communications network. The network should
More information0.0.1 Network Analysis
Graph Theory 0.0.1 Network Analysis Prototype Example: In Algonquian Park the rangers have set up snowmobile trails with various stops along the way. The system of trails is our Network. The main entrance
More informationGraph Connectivity G G G
Graph Connectivity 1 Introduction We have seen that trees are minimally connected graphs, i.e., deleting any edge of the tree gives us a disconnected graph. What makes trees so susceptible to edge deletions?
More information6. Lecture notes on matroid intersection
Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans May 2, 2017 6. Lecture notes on matroid intersection One nice feature about matroids is that a simple greedy algorithm
More informationCHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism
CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition 1.1.1. A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented
More informationCHAPTER 10 GRAPHS AND TREES. Alessandro Artale UniBZ - artale/z
CHAPTER 10 GRAPHS AND TREES Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/z SECTION 10.1 Graphs: Definitions and Basic Properties Copyright Cengage Learning. All rights reserved. Graphs: Definitions
More informationv V Question: How many edges are there in a graph with 10 vertices each of degree 6?
ECS20 Handout Graphs and Trees March 4, 2015 (updated 3/9) Notion of a graph 1. A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of pairs of elements of V called edges.
More informationLesson 22: Basic Graph Concepts
Lesson 22: asic Graph oncepts msc 175 iscrete Mathematics 1. Introduction graph is a mathematical object that is used to model different relations between objects and processes: Linked list Flowchart of
More informationGraphs II: Trailblazing
Graphs II: Trailblazing 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,, e n of the graph such that f(e 1 )={x 0,x 1 }, f(e 2 )={x
More informationGraph Theory. Part of Texas Counties.
Graph Theory Part of Texas Counties. We would like to visit each of the above counties, crossing each county only once, starting from Harris county. Is this possible? This problem can be modeled as a graph.
More informationCSE 101, Winter Discussion Section Week 1. January 8 - January 15
CSE 101, Winter 2018 Discussion Section Week 1 January 8 - January 15 Important Annotations were added (post-lecture) to the tablet slides, to fill in a few gaps (Lecture 1) Look through Additional Resources
More informationSome Strong Connectivity Concepts in Weighted Graphs
Annals of Pure and Applied Mathematics Vol. 16, No. 1, 2018, 37-46 ISSN: 2279-087X (P), 2279-0888(online) Published on 1 January 2018 www.researchmathsci.org DOI: http://dx.doi.org/10.22457/apam.v16n1a5
More informationWorksheet for the Final Exam - Part I. Graphs
Worksheet for the Final Exam - Part I. Graphs Date and Time: May 10 2012 Thursday 11:50AM~1:50PM Location: Eng 120 Start with the Self-Test Exercises (pp.816) in Prichard. 1. Give the adjacency matrix
More informationSuper Connectivity of Line Graphs and Digraphs
Acta Mathematicae Applicatae Sinica, English Series Vol. 22, No. 1 (2006) 43 48 Super Connectivity of Line Graphs and Digraphs Min Lü 1, Jun-Ming Xu 2 1 Department of Computer Science and Technology, University
More informationAMS /672: Graph Theory Homework Problems - Week V. Problems to be handed in on Wednesday, March 2: 6, 8, 9, 11, 12.
AMS 550.47/67: Graph Theory Homework Problems - Week V Problems to be handed in on Wednesday, March : 6, 8, 9,,.. Assignment Problem. Suppose we have a set {J, J,..., J r } of r jobs to be filled by a
More information11/26/17. directed graphs. CS 220: Discrete Structures and their Applications. graphs zybooks chapter 10. undirected graphs
directed graphs S 220: iscrete Structures and their pplications collection of vertices and directed edges graphs zybooks chapter 10 collection of vertices and edges undirected graphs What can this represent?
More informationGraphs and Network Flows IE411. Lecture 21. Dr. Ted Ralphs
Graphs and Network Flows IE411 Lecture 21 Dr. Ted Ralphs IE411 Lecture 21 1 Combinatorial Optimization and Network Flows In general, most combinatorial optimization and integer programming problems are
More informationDefinition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1
Graph fundamentals Bipartite graph characterization Lemma. If a graph contains an odd closed walk, then it contains an odd cycle. Proof strategy: Consider a shortest closed odd walk W. If W is not a cycle,
More informationCuts, Connectivity, and Flow
Cuts, Connectivity, and Flow Vertex Cut and Connectivity A separating set or vertex cut of a graph G is a set S V(G) such that G S G S has more than one component A graph G is k-connected if every vertex
More informationDefinition: A graph G = (V, E) is called a tree if G is connected and acyclic. The following theorem captures many important facts about trees.
Tree 1. Trees and their Properties. Spanning trees 3. Minimum Spanning Trees 4. Applications of Minimum Spanning Trees 5. Minimum Spanning Tree Algorithms 1.1 Properties of Trees: Definition: A graph G
More informationMath Summer 2012
Math 481 - Summer 2012 Final Exam You have one hour and fifty minutes to complete this exam. You are not allowed to use any electronic device. Be sure to give reasonable justification to all your answers.
More informationNetwork Topology and Graph
Network Topology Network Topology and Graph EEE442 Computer Method in Power System Analysis Any lumped network obeys 3 basic laws KVL KCL linear algebraic constraints Ohm s law Anawach Sangswang Dept.
More informationTheorem 2.9: nearest addition algorithm
There are severe limits on our ability to compute near-optimal tours It is NP-complete to decide whether a given undirected =(,)has a Hamiltonian cycle An approximation algorithm for the TSP can be used
More informationDecreasing a key FIB-HEAP-DECREASE-KEY(,, ) 3.. NIL. 2. error new key is greater than current key 6. CASCADING-CUT(, )
Decreasing a key FIB-HEAP-DECREASE-KEY(,, ) 1. if >. 2. error new key is greater than current key 3.. 4.. 5. if NIL and.
More information