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?

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.