1 Euler Circuits. Many of the things we do in daily life involve networks, like the one shown below [Houston Street Map]. Figure 1: Houston City Map
|
|
- Erik Daniels
- 5 years ago
- Views:
Transcription
1 1 Euler Circuits Many of the things we do in daily life involve networks, like the one shown below [Houston Street Map]. Figure 1: Houston City Map Suppose you needed to vist a number of locations in the Houston area, by car. In order to be more efficient, you might wish to minimize the total distance travelled minimize the total time travelled minimize the total cost (including tolls) To examine this more closely, we need to introduce some concepts:
2 Graph Vertex Edge Path Circuit Example: Parking Meters (from book)
3 A more complicated example is scheduling air plane flight paths connecting 5 cities: New York, London, Berlin, Miami and Rome. A special type of circuit is called an Euler Circuit:
4 A more realistic problem is shown below: Figure 2: Snapshot of Airplane Traffice over College Station
5 As another simpler example, suppose you need to visit a number of places on campus, by foot. Figure 3: Texas A&M campus Map What is the best path? How do you determine the best strategy? The answer to this involves an area of mathematics called operations research.
6 2 Finding Euler Circuits There are two basic questions we need to consider: Q1: Given a graph (network) G, Does an Euler circuit exist? Q2: If it does, how do we find it? We could always do trial and error, but we want to develop a more systematic (mathematical) approach. One which will apply to any graph! Euler (1735) considered a very famous problem, the Bridges of Koenigsberg, and came up with a solution based on the following ideas Valence Connectedness He eventually arrived at the famous Euler Circuit Theorem: If G is graph, which is connected and all valences are even, then an Euler circuit exists! If G has an Euler circuit, then G is connected and all valances are even!
7 How do we construct an Euler circuit? Rule: Don t split the graph! That is, don t use an edge which is the only connection between two parts of the graph which need to be covered. Don t burn your bridges after you cross them
8 Why is the Euler Circuit Theorem true? Suppose G has an Euler Circuit R, look at the starting (and ending) point X. It pairs up two edges. In fact every incoming edge is paired with an outgoing edge at X. If X is not a starting/ending point we can still pair up incoming and outgoing edges. Every point has paired edges, therefore valence is even! To show it is connected, we note that every point has at least two edges (no point has valence zero). If the graph were not connected, e.g. below then there would not be a path connecting all the points, which contradicts the fact that the graph is Euler.
9 3 Eulerizing Graphs If G doesn t have an Euler circuit (by having odd valences), can we add edges to make an Euler Circuit? The answer is yes! For example: This process is called Eulerizing a graph. A natural question is Given a non-euler graph, Is there a minimum number of edges that can be added to make it Euler? This minimizes the cost of making a graph an Euler graph. This problem often comes up in designing postal routes. This is called the Chinese Postman problem, because it was first studied by a Chinese mathematician (Meigu Guan in 1962). There is more than one way to Eulerize a graph, for example, consider the simple network below: On the left, we add two edges. On the right we only add one edge. We can squeeze them down (to make a two way street out of a one-way street). We must be careful when we compute the valences...
10 There are some rules of thumb which allow one to construct nearly optimal edge constructions. I. For a rectangular network (streets align on a grid), we can use edge-walking. Since the odd valences are on the boundary, we can walk along the boundary and connect an odd valene vertex with the next vertex. If it is even we skip it, if it is odd, we repeat the process. II. For non-rectangular networks, we connect nearest odd-valence vertices by a path (shortest length) and double the edges along it. Tthe number of edges we add is at least as great as the number of odd-valence vertices divided by two! 4 Urban Graph Problems In a city or town, the networks associated with streets also have various services associaed with them (sewers, utility poles, mailboxes, parking, etc). These have costs associated with them. Minimizing costs for services based on an analysis of urban networks can potentially save quite a bit of money [see text page 18].
The Bridges of Konigsberg Problem Can you walk around the town crossing each bridge only once?
The Bridges of Konigsberg Problem Can you walk around the town crossing each bridge only once? Many people had tried the walk and felt that it was impossible, but no one knew for sure. In 1736, Leonard
More informationChinese Postman Problem
Special Topics Chinese Postman Problem In real life, not all problems will be perfect Euler Circuits. If no Euler Circuit exists (there are odd valences present), you want to minimize the length of the
More informationChapter 1 Urban Services
Chapter 1 Urban Services For All Practical Purposes: Effective Teaching At the beginning of the course, go out of your way to learn students names. Naturally if you have a very large class this would be
More informationChapter 5: Euler Paths and Circuits The Mathematics of Getting Around
1 Finite Math A Chapter 5: Euler Paths and Circuits The Mathematics of Getting Around Academic Standards Covered in this Chapter: *************************************************************************************
More informationChapter 5: Euler Paths and Circuits The Mathematics of Getting Around
1 Finite Math A Chapter 5: Euler Paths and Circuits The Mathematics of Getting Around Academic Standards Covered in this Chapter: *************************************************************************************
More informationNetworks and Graphs: Circuits, Paths, and Graph Structures VII.A Student Activity Sheet 1: Euler Circuits and Paths
The Königsberg Bridge Problem The following figure shows the rivers and bridges of Königsberg. Residents of the city occupied themselves by trying to find a walking path through the city that began and
More informationNetworks and Graphs: Circuits, Paths, and Graph Structures VII.A Student Activity Sheet 1: Euler Circuits and Paths
The Königsberg Bridge Problem The following figure shows the rivers and bridges of Königsberg. Residents of the city occupied themselves by trying to find a walking path through the city that began and
More informationMath 167 Review 1 (c) Janice Epstein
Math 167 Review 1 (c) Janice Epstein HAPTER 1 URBAN SERVIES A graph is a collection of one or more points (vertices). The vertices may be connected by edges. Two vertices are adjacent if they are connected
More informationRosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples
Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 9.5 Euler and Hamilton Paths Page references correspond to locations of Extra Examples icons in the textbook. p.634,
More informationChapter 1. Urban Services. Chapter Outline. Chapter Summary
Chapter 1 Urban Services Chapter Outline Introduction Section 1.1 Euler Circuits Section 1.2 Finding Euler Circuits Section 1.3 Beyond Euler Circuits Section 1.4 Urban Graph Traversal Problems Chapter
More informationEULERIAN GRAPHS AND ITS APPLICATIONS
EULERIAN GRAPHS AND ITS APPLICATIONS Aruna R 1, Madhu N.R 2 & Shashidhar S.N 3 1.2&3 Assistant Professor, Department of Mathematics. R.L.Jalappa Institute of Technology, Doddaballapur, B lore Rural Dist
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 informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Test Review Ch 4 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) According to the 2000 U. S. Census, 6.753% of the U.S. population
More informationCircuits and Paths. April 13, 2014
Circuits and Paths April 13, 2014 Warm Up Problem Quandroland is an insect country that has four cities. Draw all possible ways tunnels can join the cities in Quadroland. (Remember that some cities might
More informationMathematical Thinking. Chapter 1 Graphs and Euler Circuits
Mathematical Thinking Chapter 1 Graphs and Euler Circuits Management Science A branch of applied mathematics dedicated to solving complex optimization problems. In an optimization problem, the goal is
More informationJunior Circle Meeting 3 Circuits and Paths. April 18, 2010
Junior Circle Meeting 3 Circuits and Paths April 18, 2010 We have talked about insect worlds which consist of cities connected by tunnels. Here is an example of an insect world (Antland) which we saw last
More information11.2 Eulerian Trails
11.2 Eulerian Trails K.. onigsberg, 1736 Graph Representation A B C D Do You Remember... Definition A u v trail is a u v walk where no edge is repeated. Do You Remember... Definition A u v trail is a u
More information1. The Highway Inspector s Problem
MATH 100 Survey of Mathematics Fall 2009 1. The Highway Inspector s Problem The Königsberg Bridges over the river Pregel C c d e A g D a B b Figure 1. Bridges f Is there a path that crosses every bridge
More informationEuler and Hamilton paths. Jorge A. Cobb The University of Texas at Dallas
Euler and Hamilton paths Jorge A. Cobb The University of Texas at Dallas 1 Paths and the adjacency matrix The powers of the adjacency matrix A r (with normal, not boolean multiplication) contain the number
More informationGraph Theory. 26 March Graph Theory 26 March /29
Graph Theory 26 March 2012 Graph Theory 26 March 2012 1/29 Graph theory was invented by a mathematician named Euler in the 18th century. We will see some of the problems which motivated its study. However,
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 informationGraph theory was invented by a mathematician named Euler in the 18th century. We will see some of the problems which motivated its study.
Graph Theory Graph theory was invented by a mathematician named Euler in the 18th century. We will see some of the problems which motivated its study. However, it wasn t studied too systematically until
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 informationGraph Theory. 1 Introduction to Graphs. Martin Stynes Department of Mathematics, UCC January 26, 2011
Graph Theory Martin Stynes Department of Mathematics, UCC email: m.stynes@ucc.ie January 26, 2011 1 Introduction to Graphs 1 A graph G = (V, E) is a non-empty set of nodes or vertices V and a (possibly
More informationChapter 5: The Mathematics of Getting Around
Euler Paths and Circuits Chapter 5: The Mathematics of Getting Around 5.1 Street-Routing Problem Our story begins in the 1700s in the medieval town of Königsberg, in Eastern Europe. At the time, Königsberg
More informationMEI Further Mathematics Support Programme
Further Mathematics Support Programme the Further Mathematics Support Programme www.furthermaths.org.uk Modelling and problem solving with Networks Sharon Tripconey Let Maths take you Further Nov 2009
More information8. The Postman Problems
8. The Postman Problems The Chinese postman problem (CPP) A postal carrier must pick up the mail at the post office, deliver the mail along blocks on the route, and finally return to the post office. To
More informationSalvador Sanabria History of Mathematics. Königsberg Bridge Problem
Salvador Sanabria History of Mathematics Königsberg Bridge Problem The Problem of the Königsberg Bridge There is a famous story from Konigsberg. The city of Konigsberg, Northern Germany has a significant
More informationStreet-Routing Problems
Street-Routing Problems Lecture 26 Sections 5.1-5.2 Robb T. Koether Hampden-Sydney College Wed, Oct 25, 2017 Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 1 / 21 1
More informationIE 102 Spring Routing Through Networks - 1
IE 102 Spring 2017 Routing Through Networks - 1 The Bridges of Koenigsberg: Euler 1735 Graph Theory began in 1735 Leonard Eüler Visited Koenigsberg People wondered whether it is possible to take a walk,
More information14.2 Euler Paths and Circuits filled in.notebook November 18, Euler Paths and Euler Circuits
14.2 Euler Paths and Euler Circuits The study of graph theory can be traced back to the eighteenth century when the people of the town of Konigsberg sought a solution to a popular problem. They had sections
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 informationWorksheet 28: Wednesday November 18 Euler and Topology
Worksheet 28: Wednesday November 18 Euler and Topology The Konigsberg Problem: The Foundation of Topology The Konigsberg Bridge Problem is a very famous problem solved by Euler in 1735. The process he
More informationTopic 10 Part 2 [474 marks]
Topic Part 2 [474 marks] The complete graph H has the following cost adjacency matrix Consider the travelling salesman problem for H a By first finding a minimum spanning tree on the subgraph of H formed
More informationMajority and Friendship Paradoxes
Majority and Friendship Paradoxes Majority Paradox Example: Small town is considering a bond initiative in an upcoming election. Some residents are in favor, some are against. Consider a poll asking the
More informationStudy Guide Mods: Date:
Graph Theory Name: Study Guide Mods: Date: Define each of the following. It may be helpful to draw examples that illustrate the vocab word and/or counterexamples to define the word. 1. Graph ~ 2. Vertex
More information08. First and second degree equations
08. First and second degree equations GRAPH THEORY Based on Chris K. Caldwell work: http://primes.utm.edu/cgi-bin/caldwell/tutor/graph/index.html INTRODUCTION Consider the next problem: Old Königsberg
More informationThe Traveling Salesman Problem
The Traveling Salesman Problem Hamilton path A path that visits each vertex of the graph once and only once. Hamilton circuit A circuit that visits each vertex of the graph once and only once (at the end,
More informationClassic Graph Theory Problems
Classic Graph Theory Problems Hiroki Sayama sayama@binghamton.edu The Origin Königsberg bridge problem Pregel River (Solved negatively by Euler in 176) Representation in a graph Can all the seven edges
More information11-5 Networks. Königsberg Bridge Problem
Section 11-5 Networks 1 11-5 Networks In the 1700s, the people of Königsberg, Germany (now Kaliningrad in Russia), used to enjoy walking over the bridges of the Pregel River. There were three landmasses
More information8.2 Paths and Cycles
8.2 Paths and Cycles Degree a b c d e f Definition The degree of a vertex is the number of edges incident to it. A loop contributes 2 to the degree of the vertex. (G) is the maximum degree of G. δ(g) is
More informationINTRODUCTION TO GRAPH THEORY. 1. Definitions
INTRODUCTION TO GRAPH THEORY D. JAKOBSON 1. Definitions A graph G consists of vertices {v 1, v 2,..., v n } and edges {e 1, e 2,..., e m } connecting pairs of vertices. An edge e = (uv) is incident with
More informationGRAPH THEORY AND LOGISTICS
GRAPH THEORY AND LOGISTICS Maja Fošner and Tomaž Kramberger University of Maribor Faculty of Logistics Mariborska cesta 2 3000 Celje Slovenia maja.fosner@uni-mb.si tomaz.kramberger@uni-mb.si Abstract This
More information5.5 The Travelling Salesman Problem
0 Matchings and Independent Sets 5.5 The Travelling Salesman Problem The Travelling Salesman Problem A travelling salesman, starting in his own town, has to visit each of towns where he should go to precisely
More informationUrban Geometry The tradition of mathematical playfulness set in motion by Euler is still alive well today...
1 of 9 2011/04/20 16:37 Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS Urban Geometry The tradition of mathematical playfulness set
More informationEECS 203 Lecture 20. More Graphs
EECS 203 Lecture 20 More Graphs Admin stuffs Last homework due today Office hour changes starting Friday (also in Piazza) Friday 6/17: 2-5 Mark in his office. Sunday 6/19: 2-5 Jasmine in the UGLI. Monday
More informationLaunch problem: Lining streets
Math 5340 June 15,2012 Dr. Cordero Launch problem: Lining streets Lining Street Problem A Problem on Eulerian Circuits http://www.edmath.org/mattours/discrete/ Your job for the day is to drive slowly around
More information14 Graph Theory. Exercise Set 14-1
14 Graph Theory Exercise Set 14-1 1. A graph in this chapter consists of vertices and edges. In previous chapters the term was used as a visual picture of a set of ordered pairs defined by a relation or
More informationGraph Theory
Graph Theory 2012.04.18 Our goal today is to learn some basic concepts in graph theory and explore fun problems using graph theory. A graph is a mathematical object that captures the notion of connection.
More informationFigure 2.1: A bipartite graph.
Matching problems The dance-class problem. A group of boys and girls, with just as many boys as girls, want to dance together; hence, they have to be matched in couples. Each boy prefers to dance with
More informationSimple Graph. General Graph
Graph Theory A graph is a collection of points (also called vertices) and lines (also called edges), with each edge ending at a vertex In general, it is allowed for more than one edge to have the same
More informationInstant Insanity Instructor s Guide Make-it and Take-it Kit for AMTNYS 2006
Instant Insanity Instructor s Guide Make-it and Take-it Kit for AMTNYS 2006 THE KIT: This kit contains materials for two Instant Insanity games, a student activity sheet with answer key and this instructor
More informationMA 111 Review for Exam 3
MA 111 Review for Exam 3 Exam 3 (given in class on Tuesday, March 27, 2012) will cover Chapter 5. You should: know what a graph is and how to use graphs to model geographic relationships. know how to describe
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 informationNotebook Assignments
Notebook Assignments These six assignments are a notebook using techniques from class in the single concrete context of graph theory. This is supplemental to your usual assignments, and is designed for
More informationLecture 1: An Introduction to Graph Theory
Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 1: An Introduction to Graph Theory Week 1 Mathcamp 2011 Mathematicians like to use graphs to describe lots of different things. Groups,
More informationSEVENTH EDITION and EXPANDED SEVENTH EDITION
SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 14-1 Chapter 14 Graph Theory 14.1 Graphs, Paths and Circuits Definitions A graph is a finite set of points called vertices (singular form is vertex) connected
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 informationDiscrete mathematics
Discrete mathematics Petr Kovář petr.kovar@vsb.cz VŠB Technical University of Ostrava DiM 470-2301/02, Winter term 2018/2019 About this file This file is meant to be a guideline for the lecturer. Many
More informationTWO CONTRIBUTIONS OF EULER
TWO CONTRIBUTIONS OF EULER SIEMION FAJTLOWICZ. MATH 4315 Eulerian Tours. Although some mathematical problems which now can be thought of as graph-theoretical, go back to the times of Euclid, the invention
More information3 Euler Tours, Hamilton Cycles, and Their Applications
3 Euler Tours, Hamilton Cycles, and Their Applications 3.1 Euler Tours and Applications 3.1.1 Euler tours Carefully review the definition of (closed) walks, trails, and paths from Section 1... Definition
More informationChapter 14 Section 3 - Slide 1
AND Chapter 14 Section 3 - Slide 1 Chapter 14 Graph Theory Chapter 14 Section 3 - Slide WHAT YOU WILL LEARN Graphs, paths and circuits The Königsberg bridge problem Euler paths and Euler circuits Hamilton
More informationSection 3.4 Basic Results of Graph Theory
1 Basic Results of Graph Theory Section 3.4 Basic Results of Graph Theory Purpose of Section: To formally introduce the symmetric relation of a (undirected) graph. We introduce such topics as Euler Tours,
More informationCharacterization of Graphs with Eulerian Circuits
Eulerian Circuits 3. 73 Characterization of Graphs with Eulerian Circuits There is a simple way to determine if a graph has an Eulerian circuit. Theorems 3.. and 3..2: Let G be a pseudograph that is connected
More informationSarah Will Math 490 December 2, 2009
Sarah Will Math 490 December 2, 2009 Euler Circuits INTRODUCTION Euler wrote the first paper on graph theory. It was a study and proof that it was impossible to cross the seven bridges of Königsberg once
More information7 Next choice is 4, not 3...
Math 0 opics for rst exam Chapter 1: Street Networks Operations research Goal: determine how to carry out a sequence of tasks most eciently Ecient: least cost, least time, least distance,... Example: reading
More informationDesign and Analysis of Algorithms
CS4335: Design and Analysis of Algorithms Who we are: Dr. Lusheng WANG Dept. of Computer Science office: B6422 phone: 2788 9820 e-mail: lwang@cs.cityu.edu.hk Course web site: http://www.cs.cityu.edu.hk/~lwang/ccs3335.html
More informationSection Graphs, Paths, and Circuits. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 14.1 Graphs, Paths, and Circuits INB Table of Contents Date Topic Page # January 27, 2014 Test #1 14 January 27, 2014 Test # 1 Corrections 15 January 27, 2014 Section 14.1 Examples 16 January 27,
More informationIntroduction to Networks
LESSON 1 Introduction to Networks Exploratory Challenge 1 One classic math puzzle is the Seven Bridges of Königsberg problem which laid the foundation for networks and graph theory. In the 18th century
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 informationGraph Theory Mini-course
Graph Theory Mini-course Anthony Varilly PROMYS, Boston University, Boston, MA 02215 Abstract Intuitively speaking, a graph is a collection of dots and lines joining some of these dots. Many problems in
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 informationUndirected Network Summary
Undirected Network Summary Notice that the network above has multiple edges joining nodes a to d and the network has a loop at node d. Also c is called an isolated node as it is not connected to any other
More informationAQR UNIT 7. Circuits, Paths, and Graph Structures. Packet #
AQR UNIT 7 NETWORKS AND GRAPHS: Circuits, Paths, and Graph Structures Packet # BY: Introduction to Networks and Graphs: Try drawing a path for a person to walk through each door exactly once without going
More informationWarm -Up. 1. Draw a connected graph with 4 vertices and 7 edges. What is the sum of the degrees of all the vertices?
Warm -Up 1. Draw a connected graph with 4 vertices and 7 edges. What is the sum of the degrees of all the vertices? 1. Is this graph a. traceable? b. Eulerian? 3. Eulerize this graph. Warm-Up Eulerize
More information14 More Graphs: Euler Tours and Hamilton Cycles
14 More Graphs: Euler Tours and Hamilton Cycles 14.1 Degrees The degree of a vertex is the number of edges coming out of it. The following is sometimes called the First Theorem of Graph Theory : Lemma
More informationGrades 7 & 8, Math Circles 31 October/1/2 November, Graph Theory
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grades 7 & 8, Math Circles 31 October/1/2 November, 2017 Graph Theory Introduction Graph Theory is the
More informationIntroduction to Graphs
Introduction to Graphs Slides by Lap Chi Lau The Chinese University of Hong Kong This Lecture In this part we will study some basic graph theory. Graph is a useful concept to model many problems in computer
More informationChapter 5: The Mathematics of Getting Around. 5.4 Eulerizing and Semi to Eulerizing Graphs
Chapter 5: The Mathematics of Getting Around 5.4 Eulerizing and Semi to Eulerizing Graphs Bell Work Determine if each has an Euler path or an Euler circuit, then describe an example of one. 1. 2. Copyright
More informationUTILIZATION OF GIS AND GRAPH THEORY FOR DETERMINATION OF OPTIMAL MAILING ROUTE *
UTILIZATION OF GIS AND GRAPH THEORY FOR DETERMINATION OF OPTIMAL MAILING ROUTE * Dr. Mufid Abudiab, Michael Starek, Rene Lumampao, and An Nguyen Texas A&M University-Corpus Christi 1500 Ocean Drive Corpus
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 13. An Introduction to Graphs
CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 13 An Introduction to Graphs Formulating a simple, precise specification of a computational problem is often a prerequisite to writing a
More informationAn Algorithmic Approach to Graph Theory Neetu Rawat
An Algorithmic Approach to Graph Theory Neetu Rawat nrwt12345@gmail.com, Assistant Professor, Chameli Devi Group of Institutions, Indore. India. Abstract This paper compares two different minimum spanning
More informationMATH 113 Section 9.2: Topology
MATH 113 Section 9.2: Topology Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2007 Outline 1 Introduction to Topology 2 Topology and Childrens Drawings 3 Networks 4 Conclusion Geometric Topology
More informationGraphs. Reading Assignment. Mandatory: Chapter 3 Sections 3.1 & 3.2. Peeking into Computer Science. Jalal Kawash 2010
Graphs Mandatory: hapter 3 Sections 3.1 & 3.2 Reading ssignment 2 Graphs bstraction of ata 3 t the end of this section, you will be able to: 1.efine directed and undirected graphs 2.Use graphs to model
More informationConstructing arbitrarily large graphs with a specified number of Hamiltonian cycles
Electronic Journal of Graph Theory and Applications 4 (1) (2016), 18 25 Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles Michael School of Computer Science, Engineering
More informationSolutions to the Second Midterm Exam, Math 170, Section 002 Spring 2012
Solutions to the Second Midterm Exam, Math 170, Section 002 Spring 2012 Multiple choice questions. Question 1. Suppose we have a rectangle with one side of length 5 and a diagonal of length 13. What is
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 informationGRAPH THEORY AND COMBINATORICS ( Common to CSE and ISE ) UNIT 1
GRAPH THEORY AND COMBINATORICS ( Common to CSE and ISE ) Sub code : 06CS42 UNIT 1 Introduction to Graph Theory : Definition and Examples Subgraphs Complements, and Graph Isomorphism Vertex Degree, Euler
More informationChapter 8 Topics in Graph Theory
Chapter 8 Topics in Graph Theory Chapter 8: Topics in Graph Theory Section 8.1: Examples {1,2,3} Section 8.2: Examples {1,2,4} Section 8.3: Examples {1} 8.1 Graphs Graph A graph G consists of a finite
More informationGraph Theory. Chapter 4.
Graph Theory. Chapter 4. Wandering. Here is an algorithm, due to Tarry, that constructs a walk in a connected graph, starting at any vertex v 0, traversing each edge exactly once in each direction, and
More informationHow can we lay cable at minimum cost to make every telephone reachable from every other? What is the fastest route between two given cities?
1 Introduction Graph theory is one of the most in-demand (i.e. profitable) and heavily-studied areas of applied mathematics and theoretical computer science. May graph theory questions are applied in this
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 informationChapter 11: Graphs and Trees. March 23, 2008
Chapter 11: Graphs and Trees March 23, 2008 Outline 1 11.1 Graphs: An Introduction 2 11.2 Paths and Circuits 3 11.3 Matrix Representations of Graphs 4 11.5 Trees Graphs: Basic Definitions Informally, a
More informationNumber Theory and Graph Theory
1 Number Theory and Graph Theory Chapter 7 Graph properties By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 2 Module-2: Eulerian
More informationWhat Time Where Muddy City 10 min MSTLessonPlan.docx MSTWorksheets.pptx Discussion 5 min MSTLessonPlan.docx
MST Lesson Plan Overview Minimal Spanning Trees Summary Many networks link our society: telephone networks, utility supply networks, computer networks, and road networks. For a particular network there
More information14.1 and 14.2.notebook. March 07, Module 14 lessons 1: Distance on a coordinate Plane Lesson 2: Polygons in the Coordinate Plane
Module 14 lessons 1: Distance on a coordinate Plane Lesson 2: Reflection: a transformation of a figure that flips across a line Objectives: solve problems by graphing using coordinates and absolute value
More informationDisplaying Data with Graphs. Chapter 6 Mathematics of Data Management (Nelson) MDM 4U
Displaying Data with Graphs Chapter 6 Mathematics of Data Management (Nelson) MDM 4U Cause and Effect Diagrams Developed by Dr. Kaoru Ishikawa in 1943 (Japan) Picture composed of lines and symbols designed
More informationDiscrete Mathematics and Probability Theory Fall 2013 Vazirani Note 7
CS 70 Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 7 An Introduction to Graphs A few centuries ago, residents of the city of Königsberg, Prussia were interested in a certain problem.
More informationWeek 10: Colouring graphs, and Euler s paths. 14 and 16 November, 2018
MA284 : Discrete Mathematics Week 10: Colouring graphs, and Euler s paths http://www.maths.nuigalway.ie/ niall/ma284/ 14 and 16 November, 2018 1 Colouring The Four Colour Theorem 2 Graph colouring Chromatic
More informationGRAPHS Lecture 19 CS2110 Spring 2013
GRAPHS Lecture 19 CS2110 Spring 2013 Announcements 2 Prelim 2: Two and a half weeks from now Tuesday, April16, 7:30-9pm, Statler Exam conflicts? We need to hear about them and can arrange a makeup It would
More information