To understand why option 'C' is the correct answer, let's first define what a non-planar graph is.

A non-planar graph is a graph that cannot be drawn on a plane without any of its edges crossing. In other words, it is not possible to represent a non-planar graph in a two-dimensional space without any edge intersections.

Now, let's analyze the given options one by one and determine if they satisfy the condition of being a non-planar graph with the minimum number of vertices.

Option A: 9 edges, 6 vertices
- In order to determine if this graph is non-planar, we can use Euler's formula: V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.
- Substituting the given values, we have 6 - 9 + F = 2. Solving this equation, we get F = 5.
- According to Euler's formula, the number of faces in a planar graph is always greater than or equal to the number of edges. In this case, F = 5, which is less than E = 9, indicating that this graph is planar. Therefore, option A is not the correct answer.

Option B: 6 edges, 4 vertices
- Using the same approach as above, we have 4 - 6 + F = 2. Solving this equation, we get F = 4.
- Again, F = 4 is less than E = 6, indicating that this graph is planar. Therefore, option B is not the correct answer.

Option C: 10 edges, 5 vertices
- Applying Euler's formula, we have 5 - 10 + F = 2. Solving this equation, we get F = 7.
- F = 7 is greater than E = 10, indicating that this graph is non-planar. Therefore, option C is the correct answer.

Option D: 9 edges, 5 vertices
- Using Euler's formula, we have 5 - 9 + F = 2. Solving this equation, we get F = 6.
- F = 6 is equal to E = 9, indicating that this graph is planar. Therefore, option D is not the correct answer.

In conclusion, option C (10 edges, 5 vertices) is the correct answer because it represents a non-planar graph with the minimum number of vertices.