https://file.aiquickdraw.com/mathbot/file/1730239376933vxtrvjdm.jpg

To determine the number of regions (or "faces") this planar graph divides the plane into, we can use **Euler's formula** for planar graphs:

\[
V - E + F = 2
\]

where:
- \( V \) is the number of vertices,
- \( E \) is the number of edges, and
- \( F \) is the number of faces.

In this graph:
1. **Count the vertices \( V \)**: There are 8 vertices.
2. **Count the edges \( E \)**: Since this is a complete graph \( K_8 \) with 8 vertices, the formula for the number of edges in a complete graph is:
   \[
   E = \frac{V \times (V - 1)}{2} = \frac{8 \times 7}{2} = 28
   \]

Now we can apply Euler’s formula to find \( F \):
\[
8 - 28 + F = 2
\]
\[
F = 22
\]

So, the graph divides the plane into **22 regions (faces)**.

Would you like further details on how Euler's formula is applied here?

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1. What is Euler’s formula in planar graph theory?
2. How do we calculate edges in a complete graph \( K_n \)?
3. What makes a graph planar?
4. How would the face count change if vertices or edges were added?
5. Can Euler’s formula be applied to non-planar graphs?

**Tip:** For any complete graph \( K_n \) to be planar, \( n \) must be 4 or less (except for very specific cases).

The following graph is planar. (Trust me.) If it is drawn with no intersecting edges, into how many regions ('faces') does it cut up the plane?

Learn how to calculate the number of regions (faces) created by a planar graph using Euler's formula. This problem focuses on a complete graph K8 with 8 vertices and 28 edges, demonstrating the application of V - E + F = 2 to find the solution. Perfect for students learning advanced graph theory concepts.

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