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Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to
- a)3
- b)4
- c)5
- d)6
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let G be a simple undirected planar graph on 10 vertices with 15 edges...
If the graph is planar, then it must follow below Euler’s Formula for planar graphs
v - e + f = 2
v is number of vertices
e is number of edges
f is number of faces including bounded and unbounded
10 - 15 + f = 2
f = 7
There is always one unbounded face, so the number of bounded faces = 6
Most Upvoted Answer
Let G be a simple undirected planar graph on 10 vertices with 15 edges...
Explanation:
Given, a simple undirected planar graph G on 10 vertices with 15 edges.
Step 1: Using Euler's formula to find the number of faces:
Euler's formula states that for any planar graph, F = E - V + 2, where F is the number of faces, E is the number of edges, and V is the number of vertices.
Substituting the given values, we get:
F = 15 - 10 + 2
F = 7
Therefore, we know that there are 7 faces in any embedding of G on the plane.
Step 2: Using the fact that G is a connected graph to find the number of unbounded faces:
Since G is a connected graph, we know that there is only one unbounded face. This is because any planar graph can be embedded on the plane in such a way that there is only one unbounded face.
Step 3: Using the fact that G is simple to find the number of edges on the boundary of each face:
Since G is simple, we know that each face is a simple cycle. Therefore, the number of edges on the boundary of each face is equal to the number of vertices in the face.
Step 4: Using the fact that G has 10 vertices to find the number of bounded faces:
Since G has 10 vertices, we know that the sum of the degrees of the vertices is 2E, where E is the number of edges. Therefore, the sum of the degrees of the vertices is 30.
Since each face is a simple cycle, the sum of the degrees of the vertices in each face is equal to the number of edges on the boundary of the face. Therefore, the sum of the degrees of the vertices in all the faces is equal to the sum of the edges on the boundary of all the faces.
Let x be the number of edges on the boundary of each bounded face. Then, we can write:
1x + 1x + ... + 1x + 2x = 30
where the first term appears 6 times (since there are 6 bounded faces) and the last term represents the unbounded face.
Solving the equation, we get:
x = 3
Therefore, each bounded face has 3 edges on its boundary, and there are 6 bounded faces in any embedding of G on the plane.
Step 5: Conclusion:
Thus, the number of bounded faces in any embedding of G on the plane is 6, which is option (D).
Given, a simple undirected planar graph G on 10 vertices with 15 edges.
Step 1: Using Euler's formula to find the number of faces:
Euler's formula states that for any planar graph, F = E - V + 2, where F is the number of faces, E is the number of edges, and V is the number of vertices.
Substituting the given values, we get:
F = 15 - 10 + 2
F = 7
Therefore, we know that there are 7 faces in any embedding of G on the plane.
Step 2: Using the fact that G is a connected graph to find the number of unbounded faces:
Since G is a connected graph, we know that there is only one unbounded face. This is because any planar graph can be embedded on the plane in such a way that there is only one unbounded face.
Step 3: Using the fact that G is simple to find the number of edges on the boundary of each face:
Since G is simple, we know that each face is a simple cycle. Therefore, the number of edges on the boundary of each face is equal to the number of vertices in the face.
Step 4: Using the fact that G has 10 vertices to find the number of bounded faces:
Since G has 10 vertices, we know that the sum of the degrees of the vertices is 2E, where E is the number of edges. Therefore, the sum of the degrees of the vertices is 30.
Since each face is a simple cycle, the sum of the degrees of the vertices in each face is equal to the number of edges on the boundary of the face. Therefore, the sum of the degrees of the vertices in all the faces is equal to the sum of the edges on the boundary of all the faces.
Let x be the number of edges on the boundary of each bounded face. Then, we can write:
1x + 1x + ... + 1x + 2x = 30
where the first term appears 6 times (since there are 6 bounded faces) and the last term represents the unbounded face.
Solving the equation, we get:
x = 3
Therefore, each bounded face has 3 edges on its boundary, and there are 6 bounded faces in any embedding of G on the plane.
Step 5: Conclusion:
Thus, the number of bounded faces in any embedding of G on the plane is 6, which is option (D).
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