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?

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).