In this graph, we are looking for a cycle, which is a closed loop in the graph where each vertex (point) is visited exactly once before returning to the starting point.

Let's examine the answer choices:

• A: G – P – C – F – P – G
  This path revisits vertex $P$ before completing the cycle, so it is not a valid cycle.

• B: F – G – P – G – H – F
  This path revisits vertex $G$ before completing the cycle, so it is also not a valid cycle.

• C: S – C – P – F – C – S
  This path visits vertices $S$, $C$, $P$, $F$, and returns to $S$, completing a cycle without revisiting any vertices prematurely. This is a valid cycle.

• D: P – F – H – G – F – P
  This path revisits vertex $F$ before completing the cycle, so it is not a valid cycle.

• E: C – F – H – G – P – C
  This path visits vertices $C$, $F$, $H$, $G$, $P$, and returns to $C$, completing a cycle. This is another valid cycle.

So, the correct answers are C and E.

Would you like further details or explanations about cycles in graphs?

Here are some follow-up questions you might find interesting:

1. What is the definition of a cycle in graph theory?
2. How do cycles relate to Eulerian and Hamiltonian paths?
3. What are the real-world applications of cycle detection in graphs?
4. What types of algorithms are used to find cycles in graphs?
5. Can a graph contain multiple distinct cycles? If so, how are they found?

Tip: A simple cycle is one where no vertex (except the starting and ending vertex) is repeated within the loop!