1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 14.1 Graphs, Paths, and Circuits

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Graphs Paths Circuits Bridges 14.1-2

3. Copyright 2013, 2010, 2007, 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 14.1-3

4. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Representing the Königsberg Bridge Problem Using the definitions of vertex and edge, represent the Königsberg bridge problem with a graph. Königsberg was situated on both banks and two islands of the Prigel River. From the figure, we see that the sections of town were connected with a series of seven bridges. 14.1-4

5. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Representing the Königsberg Bridge Problem 14.1-5

6. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Representing the Königsberg Bridge Problem The townspeople wondered if one could walk through town and cross all seven bridges without crossing any of the bridges twice. 14.1-6

7. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Representing the Königsberg Bridge Problem Solution Label each piece of land with a letter and draw edges to represent the bridges. 14.1-7

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Representing a Floor Plan The figure shows the floor plan of the kindergarten building at the Pullen Academy. Use a graph to represent the floor plan. 14.1-8

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Representing a Floor Plan Solution 14.1-9

10. Copyright 2013, 2010, 2007, 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, and a vertex with an odd number of edges connected to it is an odd vertex. 14.1-10

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Definitions In the figure, vertices A and D are even and vertices B and C are odd. 14.1-11

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Paths A path is a sequence of adjacent vertices and edges connecting them. C, D, A, B is an example of a path. 14.1-12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Paths A path does not need to include every edge and every vertex of a graph. In addition, a path could include the same vertices and the same edges several times. For example, on the next slide, we see a graph with four vertices. The path A, B, C, D, A, B, C, D, A, B, C, D, A, B, C starts at vertex A, “circles” the graph three times, and then goes through vertex B to vertex C. 14.1-13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Paths 14.1-14

15. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Circuit A circuit is a path that begins and ends at the same vertex. Path A, C, B, D, A forms a circuit. 14.1-15

16. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Connected Graph A graph is connected if, for any two vertices in the graph, there is a path that connects them. 14.1-16

17. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Disconnected Graph If a graph is not connected, it is disconnected. 14.1-17

18. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Bridge A bridge is an edge that if removed from a connected graph would create a disconnected graph. 14.1-18