Euler Paths & Euler Circuits

Slides:



Advertisements
Similar presentations
Introduction to Graph Theory Instructor: Dr. Chaudhary Department of Computer Science Millersville University Reading Assignment Chapter 1.
Advertisements

Graph-02.
Section 14.1 Intro to Graph Theory. Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all.
AMDM UNIT 7: Networks and Graphs
BY: MIKE BASHAM, Math in Scheduling. The Bridges of Konigsberg.
Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley. All rights reserved.
Euler Circuits and Paths
Koenigsberg bridge problem It is the Pregel River divided Koenigsberg into four distinct sections. Seven bridges connected the four portions of Koenigsberg.
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 5 The Mathematics of Getting Around 5.1Euler Circuit Problems 5.2What.
Section 2.1 Euler Cycles Vocabulary CYCLE – a sequence of consecutively linked edges (x 1,x2),(x2,x3),…,(x n-1,x n ) whose starting vertex is the ending.
Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20.
4/17/2017 Section 8.5 Euler & Hamilton Paths ch8.5.
Euler Circuit Chapter 5. Fleury’s Algorithm Euler’s theorems are very useful to find if a graph has an Euler circuit or an Euler path when the graph is.
MTH118 Sanchita Mal-Sarkar. Routing Problems The fundamental questions: Is there any proper route for the particular problem? If there are many possible.
Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.
Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley. All rights reserved.
Discrete Math Round, Round, Get Around… I Get Around Mathematics of Getting Around.
Can you find a way to cross every bridge only once?
Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.
Slide 14-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.
1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum.
Spring 2015 Mathematics in Management Science Euler’s Theorems Euler Circuits & Paths ECT & EPT Burning Bridges Fleury’s Algorithm.
Graph Theory Topics to be covered:
Slide Copyright © 2009 Pearson Education, Inc. AND Active Learning Lecture Slides For use with Classroom Response Systems Chapter 14 Graph Theory.
5.1  Routing Problems: planning and design of delivery routes.  Euler Circuit Problems: Type of routing problem also known as transversability problem.
(CSC 102) Lecture 29 Discrete Structures. Graphs.
Structures 7 Decision Maths: Graph Theory, Networks and Algorithms.
Euler and Hamilton Paths. Euler Paths and Circuits The Seven bridges of Königsberg a b c d A B C D.
CSNB143 – Discrete Structure Topic 9 – Graph. Learning Outcomes Student should be able to identify graphs and its components. Students should know how.
5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life.
Lecture 14: Graph Theory I Discrete Mathematical Structures: Theory and Applications.
Graphs, Paths & Circuits
Aim: What is an Euler Path and Circuit?
1.5 Graph Theory. Graph Theory The Branch of mathematics in which graphs and networks are used to solve problems.
Vertex-Edge Graphs Euler Paths Euler Circuits. The Seven Bridges of Konigsberg.
AND.
Graph theory and networks. Basic definitions  A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins.
Euler Paths and Circuits. The original problem A resident of Konigsberg wrote to Leonard Euler saying that a popular pastime for couples was to try.
© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 15 Graph Theory.
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 14.2 Euler Paths, and Euler Circuits.
Chapter 6: Graphs 6.1 Euler Circuits
Aim: Graph Theory – Paths & Circuits Course: Math Literacy Do Now: Aim: What are Circuits and Paths? Can you draw this figure without retracing any of.
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 5 The Mathematics of Getting Around 5.1Euler Circuit Problems 5.2What.
Walks, Paths and Circuits. A graph is a connected graph if it is possible to travel from one vertex to any other vertex by moving along successive edges.
Chapter 11 - Graph CSNB 143 Discrete Mathematical Structures.
1) Find and label the degree of each vertex in the graph.
Graph Theory Euler Paths and Euler Circuits. Euler Paths & Circuits Euler Paths and Euler Circuits (Euler is pronounced the same as Oiler) An Euler path.
Chapter 14 Section 3 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND.
Fleury's Algorithm Euler Circuit Algorithm
Copyright © 2009 Pearson Education, Inc. Chapter 14 Section 1 – Slide Graph Theory Graphs, Paths & Circuits.
MAT 110 Workshop Created by Michael Brown, Haden McDonald & Myra Bentley for use by the Center for Academic Support.
Excursions in Modern Mathematics Sixth Edition
Excursions in Modern Mathematics Sixth Edition
Çizge Algoritmaları.
AND.
Konigsberg’s Seven Bridges
Can you draw this picture without lifting up your pen/pencil?
Euler Paths & Euler Circuits
Introduction to Graph Theory Euler and Hamilton Paths and Circuits
Graph Theory.
4-4 Graph Theory Trees.
Excursions in Modern Mathematics Sixth Edition
Konigsberg- in days past.
Graphs, Paths & Circuits
5 The Mathematics of Getting Around
Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.
Euler Paths and Euler Circuits
Warm Up – 3/19 - Wednesday Give the vertex set. Give the edge set.
A Survey of Mathematics with Applications
5 The Mathematics of Getting Around
Presentation transcript:

Euler Paths & Euler Circuits Graph Theory Euler Paths & Euler Circuits

WHAT YOU WILL LEARN • Euler paths and Euler circuits • Fleury’s Algorithm

Definitions An Euler path is a path that passes through each edge of a graph exactly one time. An Euler circuit is a circuit that passes through each edge of a graph exactly one time. The difference between an Euler path and an Euler circuit is that an Euler circuit must start and end at the same vertex.

Examples Euler path D, E, B, C, A, B, D, C, E Euler circuit D, E, B, C, A, B, D, C, E, F, D

Example: Euler Path and Circuits For the graphs shown, determine if an Euler path, an Euler circuit, neither, or both exist. A B C D A B C D E The graph has an Euler path but it does not have an Euler circuit. One Euler path is E, C, B, E, D, B, A, D. Each path must begin or end at vertex D or E. This graph has two odd vertices. The graph has many Euler circuits, each of which is also an Euler path. This graph has no odd vertices. One example is A, D, B, C, D, B, A.

Euler’s Theorem For a connected graph, the following statements are true: 1. A graph with no odd vertices (all even vertices) has at least one Euler path, which is also an Euler circuit. An Euler circuit can be started at any vertex and it will end at the same vertex. 2. A graph with exactly two odd vertices has at least one Euler path but no Euler circuits. Each Euler path must begin at one of the two odd vertices, and it will end at the other odd vertex. 3. A graph with more than two odd vertices has neither an Euler path nor an Euler circuit.

Example: Using Euler’s Theorem Use Euler’s theorem to determine whether an Euler path or an Euler circuit exists in the figures shown from the previous example. A B C D

Example: Using Euler’s Theorem (continued) B C D The graph has no odd vertices (all vertices are even). According to item 1, at least one Euler circuit exists. An Euler circuit can be determined by starting at any vertex. The Euler circuit will end at the vertex from which it started. Remember that each Euler circuit is also an Euler path.

Example: Using Euler’s Theorem (continued) B C D E There are 3 even vertices (A, B, C) and two odd vertices (D, E). Based on item 2, we conclude that since there are exactly two odd vertices, at least one Euler path exists but no Euler circuits exist. Each Euler path must begin at one of the odd vertices and end at the other odd vertex.

Example a) Is it possible to travel among the states and cross each common state border exactly one time? b) If it is possible, can he start and end in the same state? Michigan Ohio Indiana Kentucky West Virginia

Solution MI OH IN KY WV We are looking for an Euler path, you must use each edge exactly one time. There are two odd vertices. Therefore, according to item 2, the graph has at least one Euler path but no Euler circuits. Therefore, yes, it is possible to travel among these states and cross each common border exactly one time. The researcher must start in either IN or KY and end in the other state. There is not an Euler circuit, so the researcher cannot start and end in the same state.

Fleury’s Algorithm To determine an Euler path or an Euler circuit: 1. Use Euler’s theorem to determine whether an Euler path or an Euler circuit exists. If one exists, proceed with steps 2-5. 2. If the graph has no odd vertices (therefore has an Euler circuit, which is also an Euler path), choose any vertex as the starting point. If the graph has exactly two odd vertices (therefore has only an Euler path), choose one of the two odd vertices as the starting point.

Fleury’s Algorithm (continued) 3. Begin to trace edges as you move through the graph. Number the edges as you trace them. Since you can’t trace any edges twice in Euler paths and Euler circuits, once an edge is traced consider it “invisible.” 4. When faced with a choice of edges to trace, if possible, choose an edge that is not a bridge (i.e., don’t create a disconnected graph with your choice of edges). 5. Continue until each edge of the entire graph has been traced once.

Example Use Fluery’s algorithm to determine an Euler circuit. There is at least one Euler circuit since there are no odd vertices. Start at any vertex to determine an Euler circuit. A B C D E F G

Example (continued) Start at C. B C D E F G 1 2 3 4 5 6 7 8 9 10 start here Start at C. Choose either CB or CD. Continue to trace from vertex to vertex around the outside of the graph.

In the following graph, determine an Euler circuit. a. CBAECDA b. CBAECDAC c. EABCDA d. AEABCD

In the following graph, determine an Euler circuit. a. CBAECDA b. CBAECDAC c. EABCDA d. AEABCD

Is it possible for a person to walk through each doorway in the house, whose floor plan is shown below, without using any of the doorways twice? If so, indicate which room the person may start and where he or she will end. a. Yes; A-G b. Yes; A-C c. Yes; C- G d. No

Is it possible for a person to walk through each doorway in the house, whose floor plan is shown below, without using any of the doorways twice? If so, indicate which room the person may start and where he or she will end. a. Yes; A-G b. Yes; A-C c. Yes; C- G d. No

Use Fleury’s algorithm to determine an Euler circuit in the following graph. a. BCFAEDA b. DABCFAE c. EDABCFA d. AEDABCFA

Use Fleury’s algorithm to determine an Euler circuit in the following graph. a. BCFAEDA b. DABCFAE c. EDABCFA d. AEDABCFA