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Let G be the non-planar graph with the minimum possible number of edges. Then G has
- a)9 edges and 5 vertices
- b)9 edges and 6 vertices
- c)10 edges and 5 vertices
- d)10 edges and 6 vertice
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let G be the non-planar graph with the minimum possible number of edge...
Explanation: According to Kuratowski’s Theorem, a graph is planar if and only if it does not contain any subdivisions of the graphs K5 or K3,3.
That means K5 and K3,3 are minimum non-planar graphs. These graphs have 5 vertices with 10 edges in K5 and 6 vertices with 9 edges in K3,3 graph.
So, graph K5 has minimum vertices and maximum edges than K3,3.
So, option (B) is correct.
So, graph K5 has minimum vertices and maximum edges than K3,3.
So, option (B) is correct.
Most Upvoted Answer
Let G be the non-planar graph with the minimum possible number of edge...
Explanation:
To understand why the correct answer is option 'B', we need to first understand the concept of planar graphs and non-planar graphs.
Planar Graph:
A graph is said to be planar if it can be drawn on a plane without any edges crossing each other.
Non-Planar Graph:
A graph is said to be non-planar if it cannot be drawn on a plane without any edges crossing each other.
Now, let's analyze each option one by one.
Option (a):
If G had 5 vertices and 9 edges, we can draw a planar graph with these specifications. For example, we can draw a complete graph K5 which has 5 vertices and 10 edges. If we remove one edge from K5, we get a planar graph with 5 vertices and 9 edges.
Option (c):
If G had 5 vertices and 10 edges, we can draw a planar graph with these specifications. For example, we can draw a bipartite graph K3,3 which has 6 vertices and 9 edges. If we remove one edge from K3,3, we get a planar graph with 5 vertices and 8 edges. So, this option is also not correct.
Option (d):
If G had 6 vertices and 10 edges, we can draw a planar graph with these specifications. For example, we can draw a complete graph K6 which has 6 vertices and 15 edges. If we remove 5 edges from K6, we get a planar graph with 6 vertices and 10 edges.
Option (b):
If G had 6 vertices and 9 edges, we cannot draw a planar graph with these specifications. This is because of the following theorem:
Euler's Formula for Planar Graphs:
For any connected planar graph, we have V - E + F = 2, where V is the number of vertices, E is the number of edges and F is the number of faces.
If we assume that G has 6 vertices and 9 edges, then we can use Euler's formula to find the number of faces. We have:
6 - 9 + F = 2
F = 5
So, there are 5 faces in G. However, this is not possible because the minimum number of faces in any non-planar graph is 6. Therefore, G cannot have 6 vertices and 9 edges, and the correct answer is option (b).
Conclusion:
The minimum possible number of edges in a non-planar graph with 6 vertices is 9. Therefore, option (b) is the correct answer.
To understand why the correct answer is option 'B', we need to first understand the concept of planar graphs and non-planar graphs.
Planar Graph:
A graph is said to be planar if it can be drawn on a plane without any edges crossing each other.
Non-Planar Graph:
A graph is said to be non-planar if it cannot be drawn on a plane without any edges crossing each other.
Now, let's analyze each option one by one.
Option (a):
If G had 5 vertices and 9 edges, we can draw a planar graph with these specifications. For example, we can draw a complete graph K5 which has 5 vertices and 10 edges. If we remove one edge from K5, we get a planar graph with 5 vertices and 9 edges.
Option (c):
If G had 5 vertices and 10 edges, we can draw a planar graph with these specifications. For example, we can draw a bipartite graph K3,3 which has 6 vertices and 9 edges. If we remove one edge from K3,3, we get a planar graph with 5 vertices and 8 edges. So, this option is also not correct.
Option (d):
If G had 6 vertices and 10 edges, we can draw a planar graph with these specifications. For example, we can draw a complete graph K6 which has 6 vertices and 15 edges. If we remove 5 edges from K6, we get a planar graph with 6 vertices and 10 edges.
Option (b):
If G had 6 vertices and 9 edges, we cannot draw a planar graph with these specifications. This is because of the following theorem:
Euler's Formula for Planar Graphs:
For any connected planar graph, we have V - E + F = 2, where V is the number of vertices, E is the number of edges and F is the number of faces.
If we assume that G has 6 vertices and 9 edges, then we can use Euler's formula to find the number of faces. We have:
6 - 9 + F = 2
F = 5
So, there are 5 faces in G. However, this is not possible because the minimum number of faces in any non-planar graph is 6. Therefore, G cannot have 6 vertices and 9 edges, and the correct answer is option (b).
Conclusion:
The minimum possible number of edges in a non-planar graph with 6 vertices is 9. Therefore, option (b) is the correct answer.
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Let G be the non-planar graph with the minimum possible number of edges. Then G hasa)9 edges and 5 verticesb)9 edges and 6 verticesc)10 edges and 5 verticesd)10 edges and 6 verticeCorrect answer is option 'B'. Can you explain this answer?
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