To determine if it is possible for a connected graph with 8 vertices and 10 edges to be drawn such that no edges cross and it creates 4 regions, we 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, including the outer region).

Step 1: Substitute the known values

Given:

• $V = 8$,
• $E = 10$,
• $F = 4$ (the number of regions we aim for),

Substitute into Euler's formula: $8 - 10 + F = 2$ $F = 4$

The equation holds true. So, theoretically, it is possible for this graph to satisfy Euler's formula.

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Step 2: Check for planarity

For a graph to be planar, it must satisfy the inequality:

E \leq 3V - 6 \quad \text{(for graphs with V \geq 3)}.

Substitute $V = 8$ into the inequality: $E \leq 3(8) - 6$ $E \leq 18$

Here, $E = 10$, which satisfies this condition. Therefore, the graph can be planar.

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Step 3: Confirm regions in a planar drawing

A planar graph with $V = 8$ and $E = 10$ can indeed be drawn without edge crossings and with exactly 4 regions, satisfying both Euler's formula and planarity conditions.

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Conclusion

Yes, it is possible to draw a connected graph with 8 vertices and 10 edges such that no edges cross and it creates 4 regions.

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Would you like a visual demonstration of how this graph might look or any clarification?

Here are 5 related questions you might explore:

1. What are the conditions for a graph to be planar?
2. Can a graph with more than $3V - 6$ edges ever be planar?
3. How does Euler’s formula apply to non-planar graphs?
4. What is the relationship between connectivity and planarity in graphs?
5. How would the graph's properties change if the number of edges increased?

Tip: When analyzing planar graphs, always check Euler's formula and the inequality $E \leq 3V - 6$ to ensure planarity.