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### Excursions in Modern MathematicsSixth Edition

### Chapter 5Euler Circuits

### Euler Circuits

### Euler Circuits

### Euler Circuits

### Euler Circuits

### Euler Circuits

### Euler Circuits

Euler CircuitsEuler Circuits### Euler Circuits

Peter Tannenbaum

The Circuit Comes to Town

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 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.

5.1 Euler Circuit Problems

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 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 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 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. .

5.2 Graphs

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 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 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 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 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 Circuits

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

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.

5.3 Graph Concepts and Terminology

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 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 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 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 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 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 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 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 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 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 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.

5.4 Graph Models

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 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 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.

5.5 Euler’s Theorems

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 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 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 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.

5.6 Fleury’s Algorithm

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 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 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 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 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.

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

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

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.]

5.7 Eulerizing Graphs

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 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 Circuits

Eulerizing Graphs

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

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 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 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 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

- 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

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