Let's solve each question based on the provided image.

Question 1 (2 Marks)

Task: Redraw the given planar graph without any intersecting edges and state the number of faces.

Solution: To redraw the graph without intersecting edges, let's perform the following steps:

1. Observe the edges that intersect and rearrange the vertices and edges to avoid these intersections while maintaining the graph's connectivity.

2. For the given graph:

   • Move vertex $E$ outside the region between vertices $A$ and $F$.
   • Adjust the edges connecting to $E$ so they no longer intersect with others.

   The graph can be redrawn as shown:

         A ---- E
         |\    / \
         | \  /   \
         |  \/     F
         |  /\     /
         | /  \   /
         |/    \ /
         C ----- D
   

   Now, all edges are planar (no intersections), forming the required planar graph.

Number of faces:

• To find the number of faces, use Euler's formula for planar graphs: $V - E + F = 2$.
  • Vertices (V): 6
  • Edges (E): 9
  • Solving for $F$: $6 - 9 + F = 2$, hence $F = 5$.

Question 2 (2 Marks)

Task (a): Explain why an Eulerian circuit is not possible.

Solution: An Eulerian circuit is a circuit that visits every edge exactly once and returns to the starting vertex. For a graph to have an Eulerian circuit, all vertices must have even degrees.

• Checking the degrees of each vertex:

  • $A$: 3 (odd)
  • $B$: 3 (odd)
  • $C$: 2 (even)
  • $D$: 3 (odd)
  • $E$: 2 (even)

  Since vertices $A$, $B$, and $D$ have odd degrees, an Eulerian circuit is not possible.

Task (b): List an Eulerian trail from the diagram.

Solution: An Eulerian trail exists if exactly two vertices have odd degrees (start at one odd degree vertex and end at the other).

Possible Eulerian trail starting at $A$:

1. $A \rightarrow B$
2. $B \rightarrow C$
3. $C \rightarrow E$
4. $E \rightarrow D$
5. $D \rightarrow A$
6. $A \rightarrow E$
7. $E \rightarrow F$
8. $F \rightarrow D$
9. $D \rightarrow B$

Question 3 (2 Marks)

Task: Identify any bridges in the graph by highlighting them.

Solution: A bridge is an edge that, if removed, would increase the number of connected components of the graph.

From the graph:

• Bridges identified:
  • $(D, E)$ and $(C, G)$

These are bridges as their removal would disconnect portions of the graph.

Would you like more details or have any questions about these solutions?

Follow-Up Questions:

1. How would you find the number of faces in a planar graph with a different number of vertices and edges?
2. What are the criteria for a graph to have an Eulerian circuit?
3. How do you identify bridges in more complex graphs?
4. What is the significance of Euler's formula in planar graphs?
5. How can we systematically redraw a graph to make it planar?

Tip: When working with Eulerian circuits or trails, always check the degrees of the vertices first to quickly determine the possibilities for circuits or trails.