In graph theory, path and circuit (also known as a cycle) are both concepts used to describe sequences of edges and vertices, but they have distinct differences:

1. Path:

   • A path is a sequence of edges that connects a sequence of distinct vertices, such that no vertex is repeated.
   • A path does not necessarily have to start and end at the same vertex.
   • Types:
     • Simple Path: A path where no vertex is visited more than once.
     • Open Path: A path where the start and end vertices are different.
     • Length of Path: The number of edges in the path.

   Example: In a graph, if you move from vertex A to vertex B, then to vertex C, and finally to vertex D, you have a path A → B → C → D.

2. Circuit (Cycle):

   • A circuit is a path that starts and ends at the same vertex, but it should not repeat any other vertex (except the starting/ending vertex).
   • It forms a closed loop where no edge or vertex (except the start/end) is repeated.
   • A circuit is also referred to as a cycle in some contexts.

   Example: In a graph, a circuit would be if you start at vertex A, move to B, C, and D, and then return to vertex A (i.e., A → B → C → D → A).

Key Differences:

• Path: A sequence of distinct vertices and edges, not necessarily forming a loop.
• Circuit: A path that forms a closed loop, where the first and last vertices are the same.

Would you like a deeper explanation of any of these concepts, or perhaps a visual example?

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Related questions to explore:

1. What is a Hamiltonian Path in graph theory?
2. What is a Eulerian Circuit and how does it differ from a regular circuit?
3. How do we define the length of a path in a graph?
4. What conditions must be met for a graph to have a Eulerian Path or Eulerian Circuit?
5. How can we prove if a graph contains a cycle or not?

Tip: When working with paths and circuits, it’s helpful to visualize them using simple graphs to understand their structure clearly!