excursions in modern mathematics sixth edition l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Excursions in Modern Mathematics Sixth Edition PowerPoint Presentation
Download Presentation
Excursions in Modern Mathematics Sixth Edition

Loading in 2 Seconds...

play fullscreen
1 / 56

Excursions in Modern Mathematics Sixth Edition - PowerPoint PPT Presentation


  • 377 Views
  • Uploaded on

Excursions in Modern Mathematics Sixth Edition. Peter Tannenbaum. Chapter 5 Euler Circuits. The Circuit Comes to Town. Euler Circuits Outline/learning Objectives. To identify and model Euler circuit and Euler path problems. To understand the meaning of basic graph terminology.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Excursions in Modern Mathematics Sixth Edition' - kolton


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
chapter 5 euler circuits

Chapter 5Euler Circuits

The Circuit Comes to Town

euler circuits outline learning objectives
Euler CircuitsOutline/learning Objectives
  • To identify and model Euler circuit and Euler path problems.
  • To understand the meaning of basic graph terminology.
  • To classify which graphs have Euler circuits or paths using Euler’s circuit theorems.
euler circuits outline learning objectives cont
Euler CircuitsOutline/learning Objectives (cont.)
  • To implement Fleury’s algorithm to find an Euler circuit or path when it exists.
  • To eulerize or semi-eulerize graphs when necessary.
  • To recognize an optimal eulerization (semi-eulerization) of a graph.
euler circuits

Euler Circuits

5.1 Euler Circuit Problems

euler circuits6
Euler Circuits

What is a routing problem?

  • Existence question

Is an actual route possible?

  • Optimization question

Of all the possible routes, which one is the optimal route?

euler circuits7
Euler Circuits

We will answer both the existence and optimization questions

for a special class of routing problems known as Euler circuit

problems. The common thread is what we call the exhaustion

requirement.

euler circuits8
Euler Circuits

The name of the game is to trace each drawing

without lifting the pencil or retracing any of the lines.

These kinds of tracings are called unicursal tracings.

euler circuits9
Euler Circuits

When we end in the same place we started, we call it

a closed unicursal tracing; when we start and end in

different places, we call it an open unicursal tracing. .

euler circuits10

Euler Circuits

5.2 Graphs

euler circuits11
Euler Circuits
  • Vertices- dots
  • Edges- lines

The edges do not have to be straight lines. But they have to connect two vertices.

  • Loop- an edge connecting a vertex back with itself

A graph is a picture consisting of:

euler circuits12
Euler Circuits

This graph has six vertices A, B, C, D, E, and F and eight edges. The edges can be described by giving the two vertices that are connected by the edge. Thus the edges are AB, AD, BB, BC, BE, CD, CD, and DE

euler circuits13
Euler Circuits

First, note that the point where edges BE and AD cross is not a vertex– it is just the crossing point of two edges. Second, that vertex F is not connected to any other vertex. Such a vertex is called an isolated vertex.

euler circuits14
Euler Circuits

Third, note that this graph has a loop, namely the edge BB. Finally, note that it is permissible to have two edges connecting the same two vertices, as in the case with C and D. When a graph has more than one edge connecting the same pair of vertices, it is said to have multiple edges.

euler circuits15
Euler Circuits

This graph is considered a single graph, even though it consists of two separate, disconnected pieces. Such graphs are called disconnected graph, and the individual pieces are called the components of the graph..

euler circuits16
Euler Circuits

A Graph with No Edges? Yes, its possible. Without edges, every vertex of the graph is an isolated vertex.

euler circuits17
Euler Circuits

Graphs

A graph is a structure that defines pairwise relationships within a set to objects. The objects are the vertices, and the pairwise relationships are the edges: X is related to Y if and only if XY is an edge.

euler circuits18

Euler Circuits

5.3 Graph Concepts and Terminology

euler circuits19
Euler Circuits

Adjacent vertices.

Two vertices are said to be adjacent if there is an edge joining them. Vertices B and E are adjacent; C and D are not. Also because of the loop at E, we can say that Vertex E is adjacent to itself.

euler circuits20
Euler Circuits

Adjacent edges.

Two edges are adjacent if they share a common vertex. AB and AD are adjacent; edges AB and DE are not.

euler circuits21
Euler Circuits

Degree of a vertex.

The degree of a vertex is the number of edges at that vertex. When there is a loop at the vertex, the loop contributes twice. The deg(A) = 3, deg(B) = 5, deg(C) = 3, deg(D) = 2, deg(E) = 4, etc.

euler circuits22
Euler Circuits

Odd and even vertices.

An odd vertex is a vertex of odd degree; an even vertex is a vertex of even degree. The graph has two even vertices (D and E) and six odd vertices (all the others).

euler circuits23
Euler Circuits

Paths.

A path is a sequence of vertices with the property that each vertex in the sequence is adjacent to the next one. The key requirement in a path is that an edge can be part of a path only once.

euler circuits24
Euler Circuits

Paths (continued).

The number of edges in the path is called the length of the path.

A, B, E, D. This is a path from vertex A to D, consisting of the edges AB, BE, and ED. The length of this path is 3.

euler circuits25
Euler Circuits

Circuits.

A circuit has the same definition as a path, but has the additional requirement that the trip starts and ends at the same vertex.

euler circuits26
Euler Circuits

Connected graphs.

A graph is connected, if given any two vertices, there is a path joining them. A graph that is not connected is said to be disconnected. A disconnected graph is made up of separate components.

euler circuits27
Euler Circuits

Bridges.

Sometimes in a connected graph there is an edge such that if we were to erase it, the graph would become disconnected—such an edge is called a bridge. BF, FG, and FH are bridges.

euler circuits28
Euler Circuits

Euler paths.

An Euler path is a path that passes through every edge of a graph once and only once. The graph shown in (a) does not have an Euler path; the graph in (b) has several Euler paths. One of them is L,A,R,D,A,R,D,L,A.

euler circuits29
Euler Circuits

Euler circuits.

An Euler circuit is a circuit that passes through every edge of a graph. One of them is L,A,R,D,A,R,D,L,A,L.

Note that if a graph has an Euler circuit it cannot have an Euler path, and vice versa.

euler circuits30

Euler Circuits

5.4 Graph Models

euler circuits31
Euler Circuits

The notion of using a mathematical concept to describe and

solve a real-life problem is called modeling. Below is an

example of how we can use graphs to model a problem.

euler circuits32
Euler Circuits

The only thing that truly matters to the solution of this

problem is the relationship between land masses (islands

and banks) and bridges. Which land masses are connected

to each other and by how many bridges?

euler circuits33
Euler Circuits

This information is captured by the red edges in (b). We

end up with the graph model shown in (c). The four vertices

of the graph represent each of the four land masses; the edges

represent the seven bridges.

euler circuits34

Euler Circuits

5.5 Euler’s Theorems

euler circuits35
Euler Circuits

Euler’s Circuit Theorem

  • If a graph is connected, and every vertex is even, then it has an Euler circuit (at least one, usually more).
  • If a graph has any odd vertices, then it does not have an Euler circuit.
euler circuits36
Euler Circuits

The graph in (a ) cannot have an Euler circuit because it is disconnected. The graph in (b) has odd vertices (C is one of them, there are others). The graph in (c) is connected and all the vertices are even. The graph does have Euler circuits.

euler circuits37
Euler Circuits

Euler’s Path Theorem

  • If a graph is connected, and has exactly two odd vertices, then it has an Euler path (at least one, usually more). Any such path must start at one of the odd vertices and end at the other one.
  • If a graph has more than two odd vertices, then it cannot have an Euler path.
euler circuits38
Euler Circuits

Euler’s Sum of Degrees Theorem

  • The sum of the degrees of all the vertices of a graph equals twice the number of edges (and therefore is an even number).
  • A graph always has an even number of odd vertices.
euler circuits39

Euler Circuits

5.6 Fleury’s Algorithm

euler circuits40
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Preliminaries. Make sure that the graph is connected and either (1) has no odd vertices (circuit), or (2) has two odd vertices (path).
euler circuits41
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Start. Choose a starting vertex. [ In case (1) this can be any vertex; in case (2) it must be one of the two odd vertices.]
euler circuits42
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Intermediate steps. At each step, if you have a choice, don’t choose a bridge of the yet-to-be-traveled part of the graph. However, if you have only one choice, take it.
euler circuits43
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Intermediate steps. At each step, if you have a choice, don’t choose a bridge of the yet-to-be-traveled part of the graph. However, if you have only one choice, take it.
euler circuits44
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Intermediate steps. At each step, if you have a choice, don’t choose a bridge of the yet-to-be-traveled part of the graph. However, if you have only one choice, take it.
euler circuits45
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Intermediate steps. At each step, if you have a choice, don’t choose a bridge of the yet-to-be-traveled part of the graph. However, if you have only one choice, take it.
euler circuits46
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • Intermediate steps. At each step, if you have a choice, don’t choose a bridge of the yet-to-be-traveled part of the graph. However, if you have only one choice, take it.
euler circuits47
Euler Circuits

Fleury’s Algorithm for Finding an Euler Circuit (Path)

  • End. When you can’t travel any more, the circuit (path) is complete. [In case (1) you will be back at the starting vertex; in case (2) you will end at the other odd vertex.]
euler circuits48

Euler Circuits

5.7 Eulerizing Graphs

euler circuits49
Euler Circuits

Eulerizing Graphs

Our first step is to identify the odd vertices. This graph has eight odd vertices (B,C,E,F,H,I,K,and L), shown in red.

euler circuits50
Euler Circuits

Eulerizing Graphs

When we add a duplicate copy of edges BC,EF,HI, and KL, we get this graph. This is the eulerized version of the original graph.

euler circuits51
Euler Circuits

Eulerizing Graphs

This graph shows the many possible Euler circuits,with the edges numbered in the order they are traveled..

euler circuits52
Euler Circuits

Eulerizing Graphs

With the four duplicate edges (BC,EF,HI,and KL) indicating the deadhead blocks where a second pass is required. The total length of this route is 28 blocks (24 blocks in the grid plus 4 deadhead blocks).

euler circuits53
Euler Circuits

Eulerizing Graphs

In some situations we need to find an exhaustive route, but there is no requirement that it be closed—the route may start and end at different points.

euler circuits54
Euler Circuits

Eulerizing Graphs

In these cases, we want to leave two odd vertices on the graph unchanged, and change the other odd vertices into even vertices.

euler circuits55
Euler Circuits

Eulerizing Graphs

This process id called a semi-eulerization of the graph. In this case the route will start at one of the two odd vertices and end at the other one.

euler circuits conclusion
Euler Circuits Conclusion
  • Concept of a graph

This idea can be traced back to Euler some 270 years ago.

  • Concept of a graph model.

We used graphs and mathematical theory of graphs to solve certain types of routing problems.

  • Concept of an algorithm

A set of procedural rules that helps us find Euler circuits or Euler path in a graph