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

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

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

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

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

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

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

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

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

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

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

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

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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Paths

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

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

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

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