1. Copyright © 2009 Pearson Education, Inc. Chapter 14 Section 1 – Slide 1 4-1 Graph Theory Graphs, Paths & Circuits

2. Chapter 14 Section 1 – Slide 2 Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN Graphs, paths and circuits The Königsberg bridge problem

3. Chapter 14 Section 1 – Slide 3 Copyright © 2009 Pearson Education, Inc. History  This was developed by Leonhard Euler (pronounced “oiler”) to study the Konigsberg Bridge problem.  Konigsberg was situated on both banks of the Prigel River in Eastern Prussia with a series of seven bridges connecting the banks via two islands.  The people of Konigsberg wanted to know if it was possible to cross all seven of the bridges without crossing any twice.

4. Chapter 14 Section 1 – Slide 4 Copyright © 2009 Pearson Education, Inc. Definitions A graph is a finite set of points called vertices (singular form is vertex) connected by line segments (not necessarily straight) called edges. A loop is an edge that connects a vertex to itself. A B C D Loop Edge Vertex Not a vertex

5. Chapter 14 Section 1 – Slide 5 Copyright © 2009 Pearson Education, Inc. Example: Map The map shows the states that make up part of the Midwest states from Weather Underground, Inc. Construct a graph to show the states that share a common border. Michigan Ohio Indiana Kentucky West Virginia

6. Chapter 14 Section 1 – Slide 6 Copyright © 2009 Pearson Education, Inc. Solution Each vertex will represent one of the states. If two states share a common border, connect the respective vertices with an edge.

7. Chapter 14 Section 1 – Slide 7 Copyright © 2009 Pearson Education, Inc. Solution (continued) Michigan Ohio Indiana Kentucky West Virginia MI OHIN KY WV

8. Chapter 14 Section 1 – Slide 8 Copyright © 2009 Pearson Education, Inc. Definitions The degree of a vertex is the number of edges that connect to that vertex. A vertex with an even number of edges connected to it is an even vertex. A vertex with an odd number of edges connected to it is an odd vertex. MI, OH, and WV are even vertices IN, KY are odd vertices MI OHIN KY WV

9. Chapter 14 Section 1 – Slide 9 Copyright © 2009 Pearson Education, Inc. Definitions A path is a sequence of adjacent vertices and edges connecting them. C, D, A, B is an example of a path. A circuit is a path that begins and ends at the same vertex. A, C, B, D, A forms a circuit. A B C D E A B C D E

10. Chapter 14 Section 1 – Slide 10 Copyright © 2009 Pearson Education, Inc. Definitions A graph is connected if, for any two vertices in the graph, there is a path that connects them. Examples of disconnected graphs. A B C D G H JK

11. Chapter 14 Section 1 – Slide 11 Copyright © 2009 Pearson Education, Inc. Definitions (continued) A bridge is an edge that if removed from a connected graph would create a disconnected graph. A B C D bridge G H JK

12. Chapter 14 Section 1 – Slide 12 Copyright © 2009 Pearson Education, Inc. Select the graph with six vertices, a bridge, and a loop. a. c. b. d.

13. Chapter 14 Section 1 – Slide 13 Copyright © 2009 Pearson Education, Inc. Select the graph with six vertices, a bridge, and a loop. a. c. b. d.

14. Chapter 14 Section 1 – Slide 14 Copyright © 2009 Pearson Education, Inc. Represent the floor plan below as a graph where each vertex represents a room and each edge represents a doorway between rooms.

15. Chapter 14 Section 1 – Slide 15 Copyright © 2009 Pearson Education, Inc. a. c. b. d.

16. Chapter 14 Section 1 – Slide 16 Copyright © 2009 Pearson Education, Inc. a. c. b. d.

17. Chapter 14 Section 1 – Slide 17 Copyright © 2009 Pearson Education, Inc. Draw a connected graph with all even vertices. a. c. b. d.

18. Chapter 14 Section 1 – Slide 18 Copyright © 2009 Pearson Education, Inc. Draw a connected graph with all even vertices. a. c. b. d.

19. Practice Problems

20. Chapter 14 Section 1 – Slide 20 Copyright © 2009 Pearson Education, Inc.

21. Chapter 14 Section 1 – Slide 21 Copyright © 2009 Pearson Education, Inc.