https://file.aiquickdraw.com/mathbot/file/1727059604236zi2j9a7l.jpg

The question provided asks to prove that the given graph is planar and redraw it as a planar graph, then determine the number of its faces.

The graph depicted is a **complete graph on five vertices**, commonly referred to as **\( K_5 \)**. Let's break down the steps required to answer the question:

### 1. **Determining if the Graph is Planar:**
   - **\( K_5 \)** is a well-known example in graph theory of a **non-planar** graph. By Kuratowski's Theorem, a graph is non-planar if it contains a subdivision of \( K_5 \) or \( K_{3,3} \). Since this graph is \( K_5 \) itself, it is inherently non-planar.

### 2. **Redrawing the Graph as Planar:**
   - To redraw the graph as a planar graph, you will need to remove some edges so that it no longer remains a complete graph but can be embedded on a plane without edges crossing.
   - A standard way to make \( K_5 \) planar is to remove one of its edges. For example, removing one edge between any two vertices will make the graph planar.

### 3. **Determining the Number of Faces:**
   - For a **planar graph**, we use **Euler's formula**:
     \[
     V - E + F = 2
     \]
     Where:
     - \( V \) = Number of vertices
     - \( E \) = Number of edges
     - \( F \) = Number of faces (including the outer face)
   - After removing one edge from \( K_5 \), we have:
     - \( V = 5 \)
     - \( E = 9 \) (since we removed one edge from the original 10 edges of \( K_5 \))
     - Using Euler's formula, we solve for \( F \):
     \[
     5 - 9 + F = 2 \implies F = 6
     \]
     Hence, the planar version of the graph will have **6 faces**.

### Conclusion:
- \( K_5 \) is non-planar. To make it planar, one edge needs to be removed, and in its planar form, the graph will have 6 faces.

Would you like a detailed diagram of the planar redrawing? Let me know if you have any questions!

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Here are 5 related questions:
1. What is Kuratowski's Theorem and how does it classify non-planar graphs?
2. Can you explain the concept of a graph subdivision in the context of planarity?
3. How does Euler’s formula apply to polyhedra, and how is it related to graph theory?
4. What are the properties of a complete graph and how is \( K_5 \) different from other complete graphs?
5. Can you list other common non-planar graphs besides \( K_5 \) and \( K_{3,3} \)?

**Tip:** When determining whether a graph is planar, it is useful to check if it contains subdivisions of \( K_5 \) or \( K_{3,3} \), as these are standard non-planar graphs.

Show that the graph below is planar. Redraw the graph as planar. And, determine the number of its faces.

Learn how to determine if a complete graph K5 is planar, and understand how to redraw the graph as planar by removing edges. This detailed solution also applies Euler's formula to determine the number of faces for the planar graph.

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