https://file.aiquickdraw.com/mathbot/file/1730068520965p6thnnry.jpg

This exercise involves redrawing a graph to avoid edge crossings. The current graph has four main vertices, each connected to every other vertex, forming a **complete graph** \( K_4 \). Additionally, there are two extra vertices connected to the structure, likely forming a bipartite extension to create a planar version.

### Solution Outline
To make this graph planar, we can rearrange the vertices such that edges do not intersect, following a few principles of planar graph theory. Here’s a way to do it:

1. **Separate vertices**: Place the four main vertices of \( K_4 \) in a circular arrangement. 
2. **Utilize extra vertices**: Add the two extra vertices outside this circular arrangement, connecting them without crossing.
3. **Edge distribution**: Distribute edges so they follow the circle or connect only to vertices in an organized layout, avoiding intersections.

One common layout that achieves this is the **Kuratowski's theorem** application for \( K_4 \) or placing \( K_{3,3} \) graphs in a specific arrangement, often by bending the edges around.

Would you like a visual example of the redraw or further explanation on planar graph theory? 

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Here are five follow-up questions related to planar graph theory and graph drawing:

1. What is the difference between a planar and non-planar graph?
2. How can Kuratowski’s theorem help in determining if a graph is planar?
3. What are some real-world applications of planar graphs?
4. How does Euler's formula relate to planar graphs?
5. What strategies are effective in minimizing edge crossings in large graphs?

**Tip**: In graph theory, visualizing different layouts can help in understanding edge relationships better.

Don't be cross. Here is a drawing of a graph with four vertices and an edge joining every pair of those vertices plus two additional edges and vertices. Redraw the graph so that none of the edges cross.

Learn how to redraw a graph with four vertices and edges connecting every pair, ensuring that none of the edges cross. This problem explores concepts in graph theory, including planar graphs and Kuratowski's theorem. Suitable for college-level students.

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