Excursions in Modern Mathematics Sixth Edition

1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

2. Chapter 5Euler Circuits The Circuit Comes to Town

3. 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.

4. 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. Euler Circuits 5.1 Euler Circuit Problems

6. 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?

7. 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.

8. 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.

9. 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. .

10. Euler Circuits 5.2 Graphs

11. 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:

12. 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

13. 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.

14. 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.

15. 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..

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

17. 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.

18. Euler Circuits 5.3 Graph Concepts and Terminology

19. 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.

20. 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.

21. 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.

22. 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).

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

29. 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.

30. Euler Circuits 5.4 Graph Models

31. 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.

32. 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?

33. 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.

34. Euler Circuits 5.5 Euler’s Theorems

35. 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.

36. 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.

37. 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.

38. 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.

39. Euler Circuits 5.6 Fleury’s Algorithm

40. 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).

41. 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.]

42. 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.

43. 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.

44. 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.

45. 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.

46. 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.

47. 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.]

48. Euler Circuits 5.7 Eulerizing Graphs

49. 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.

50. 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.