1. Graphs, Paths, and Circuits
Section 13.1
Graphs, Paths, and Circuits

2. What You Will Learn
Upon completion of this section, you will be able to:
Represent problems using graphs.
Understand paths, circuits, and bridges.

3. 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

4. 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.

5. Example 1: Representing the Königsberg Bridge Problem

6. 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.

7. Example 1: Representing the Königsberg Bridge Problem
Solution
Label each piece of land with a letter and draw edges to represent the bridges.

8. Example 3: Representing a Floor Plan
The figure shows the floor plan of the Phenomenal Phitness gym. Use a graph to represent the floor plan.

9. Example 3: Representing a Floor Plan
Solution

10. 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.

11. Definitions
In the figure, vertices A and D are even and vertices B and C are odd. A loop, counts as 2 edges for the vertex it is attached to as there are 2 directions that can be chosen.

12. Paths
A path is a sequence of adjacent vertices and edges connecting them.
C, D, A, B is an example of a path.

13. 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. Paths

15. Circuit A circuit is a path that begins and ends at the same vertex.
Path A, C, B, D, A forms a circuit.

16. Connected Graph
A graph is connected if, for any two vertices in the graph, there is a path that connects them.

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

18. Bridge
A bridge is an edge that, if removed from a connected graph, would create a disconnected graph.