If a planar graph has 11 vertices and four times as many edges as faces, how many faces does it have? Is there a planar graph with 13 vertices that has three times more edges than faces?
If a planar graph has 11 vertices and four times as many edges as faces, how many faces does it have? Is there a planar graph with 13 vertices that has three times more edges than faces?
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter4: Selection Structures
Section: Chapter Questions
Problem 14PP
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Transcribed Image Text:**Question:**
- If a graph is planar and has \( n \) vertices, can it have two vertices of degree \( n-1 \)? Can it have three vertices of degree \( n-1 \)? Justify your answer.
![**Question:**
If a planar graph has 11 vertices and four times as many edges as faces, how many faces does it have? Is there a planar graph with 13 vertices that has three times more edges than faces?
**Explanation:**
This question involves understanding the properties of planar graphs and using the Euler's formula for planar graphs, which is:
\[ V - E + F = 2 \]
where \( V \) represents the number of vertices, \( E \) represents the number of edges, and \( F \) represents the number of faces.
For the first part:
- Given: \( V = 11 \) and \( E = 4F \).
For the second part:
- Given: \( V = 13 \) and \( E = 3F \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ae3fb89-f8a8-4394-beb4-7c14e660d345%2Fdb2742f7-e4a4-4da8-a79b-e03fc623e11b%2Fdal5q8f_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
If a planar graph has 11 vertices and four times as many edges as faces, how many faces does it have? Is there a planar graph with 13 vertices that has three times more edges than faces?
**Explanation:**
This question involves understanding the properties of planar graphs and using the Euler's formula for planar graphs, which is:
\[ V - E + F = 2 \]
where \( V \) represents the number of vertices, \( E \) represents the number of edges, and \( F \) represents the number of faces.
For the first part:
- Given: \( V = 11 \) and \( E = 4F \).
For the second part:
- Given: \( V = 13 \) and \( E = 3F \).
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