To find the number of faces (regions) in a planar graph, 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 (regions).

Step 1: Count the Vertices and Edges

In this graph:

• Vertices (V): There are 8 vertices, numbered 1 through 8.
• Edges (E): Counting the lines between vertices in the graph, there are 18 edges.

Step 2: Use Euler's Formula

Plugging in these values: $8 - 18 + F = 2$

Simplifying: $F = 2 + 18 - 8$ $F = 12$

Answer

The number of faces (regions) the graph cuts up the plane into is 12.

Would you like further details or have any other questions?

Related Questions to Explore:

1. What happens to Euler’s formula if a graph is not planar?
2. How would the number of faces change if more edges were added to this graph?
3. Can Euler’s formula be used for graphs in three dimensions, and if so, how?
4. What are the conditions that make a graph planar?
5. How does a change in the number of vertices affect the possible configurations of faces?

Tip: Euler’s formula is a powerful tool for analyzing planar graphs, and it’s especially useful in verifying whether a drawing is indeed planar if applied correctly.