   # Dual Graph Every planar graph has what is called a dual graph. To form the dual graph, start with a planar drawing of the graph. With a different-color pen, draw a dot in each face (including the infinite face). These will be the vertices of the dual graph. Now, for every edge in the original graph, draw a new edge that crosses over it and connects the two new vertices in the faces on each side of the original edge. The resulting dual graph is always itself a planar graph. (Note that it may contain multiple edges.) The procedure is illustrated below. Draw the dual graph of each of the planar graphs below. For each of the graphs in part a, count the number of faces, edges, and vertices in the original graph and compare your results with the numbers in the dual graph. What do you notice? Explain why this will always be true. Start with your dual graph of the first graph in part a, and find its dual. What do you notice? ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584

#### Solutions

Chapter
Section ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 5.3, Problem 30ES
Textbook Problem
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## Dual Graph Every planar graph has what is called a dual graph. To form the dual graph, start with a planar drawing of the graph. With a different-color pen, draw a dot in each face (including the infinite face). These will be the vertices of the dual graph. Now, for every edge in the original graph, draw a new edge that crosses over it and connects the two new vertices in the faces on each side of the original edge. The resulting dual graph is always itself a planar graph. (Note that it may contain multiple edges.) The procedure is illustrated below. Draw the dual graph of each of the planar graphs below. For each of the graphs in part a, count the number of faces, edges, and vertices in the original graph and compare your results with the numbers in the dual graph. What do you notice? Explain why this will always be true. Start with your dual graph of the first graph in part a, and find its dual. What do you notice?

To determine

(a)

To Draw:

Duel graph for each of the given graphs.

### Explanation of Solution

Given information:

Example of a dual graph:

To determine

(b)

The number of edges, vertices and faces of the given graphs and its dual graphs.

To determine

(c)

To Draw:

Dual of the dual graph.

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