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In this exercise, you are given a graph and asked to analyze modifications made to it. The task is to compute the changes in the number of vertices, edges, and regions in each modified graph.

Let's begin by examining the modifications in each of the graphs shown:

### Original Graph (First one to the right):
- **Vertices**: 6
- **Edges**: 7
- **Regions**: 3 (Regions are enclosed areas formed by the edges.)

Now, we'll go through each modified graph and compute how these quantities change:

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### First Modified Graph (Second from the left):
- A new edge has been added (a loop at the top left vertex).
- **Vertices**: Remain the same (6).
- **Edges**: Increased by 1 (8 total).
- **Regions**: Increased by 1, as the loop creates a new enclosed region (4 total).

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### Second Modified Graph (Middle):
- A new edge has been added at the bottom right, connecting two existing vertices.
- **Vertices**: Remain the same (6).
- **Edges**: Increased by 1 (8 total).
- **Regions**: Increased by 1, creating an additional region (4 total).

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### Third Modified Graph (Second from the right):
- A new edge has been added on the top right, forming a loop.
- **Vertices**: Remain the same (6).
- **Edges**: Increased by 1 (8 total).
- **Regions**: Increased by 1, as the loop creates an additional region (4 total).

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### Fourth Modified Graph (Far right):
- A new edge has been added in the middle, creating a connection between the central vertices.
- **Vertices**: Remain the same (6).
- **Edges**: Increased by 1 (8 total).
- **Regions**: Increased by 1, forming another enclosed region (4 total).

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Would you like to go deeper into how the regions were calculated or have any additional questions?

Here are some further questions for consideration:
1. How would removing an edge affect the number of regions?
2. What happens to the Euler characteristic if we add more edges without increasing vertices?
3. How does the number of faces (regions) relate to the number of edges in a planar graph?
4. How do loops affect the calculation of regions compared to adding straight edges?
5. Can you generalize a formula for how adding an edge in a planar graph affects the regions?

**Tip:** When adding loops or edges to a graph, focus on how they divide the space to count new regions effectively.

You are given a graph and asked to compute the change in the number of vertices, edges, and regions for each of the modified graphs shown.

Analyze graph modifications and compute changes in the number of vertices, edges, and regions. Using Euler's formula for planar graphs, this exercise covers key graph theory concepts, suitable for high school students studying advanced mathematics.

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