Section 14 1 Graphs Paths and Circuits Copyright

What You Will Learn Graphs Paths Circuits Bridges 14. 1 -2 Copyright 2013, 2010,

Definitions A graph is a finite set of points called vertices (singular form is

Example 1: Representing the Königsberg Bridge Problem Using the definitions of vertex and edge,

Example 1: Representing the Königsberg Bridge Problem 14. 1 -5 Copyright 2013, 2010, 2007,

Example 1: Representing the Königsberg Bridge Problem The townspeople wondered if one could walk

Example 1: Representing the Königsberg Bridge Problem Solution Label each piece of land with

Example 3: Representing a Floor Plan The figure shows the floor plan of the

Example 3: Representing a Floor Plan Solution 14. 1 -9 Copyright 2013, 2010, 2007,

Definitions The degree of a vertex is the number of edges that connect to

Definitions In the figure, vertices A and D are even and vertices B and

Paths A path is a sequence of adjacent vertices and edges connecting them. C,

Paths A path does not need to include every edge and every vertex of

Paths 14. 1 -14 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Circuit A circuit is a path that begins and ends at the same vertex.

Connected Graph A graph is connected if, for any two vertices in the graph,

Disconnected Graph If a graph is not connected, it is disconnected. 14. 1 -17

Bridge A bridge is an edge that if removed from a connected graph would

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Definitions A graph is a finite set of points called vertices (singular form is vertex) connected by line segments (not necessarily straight) called edges. Loop A loop is an edge A B that connects a vertex to itself. Not a Edge vertex C 14. 1 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc. D Vertex

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 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 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 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 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 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 starts at vertex A, “circles” the graph three times, and then goes through vertex B to vertex C. 14. 1 -13 Copyright 2013, 2010, 2007, Pearson, Education, Inc.