Define the terms Path and circuit.

Path: In graph theory, a path is a sequence of vertices (nodes) in a graph, connected by edges, with no vertex repeated. It can be expressed mathematically as:

P = (v_1, v_2, ..., v_n)

where:

• P is the path.
• $v_1, v_2, ..., v_n$ are the vertices in the path.
• $(v_1, v_2), (v_2, v_3), ..., (v_{n-1}, v_n)$ are the edges in the path.

Circuit: A circuit is a closed path in a graph, meaning it starts and ends at the same vertex. It can be expressed mathematically as:

C = (v_1, v_2, ..., v_n, v_1)

where:

• C is the circuit.
• $v_1, v_2, ..., v_n$ are the vertices in the circuit.
• $(v_1, v_2), (v_2, v_3), ..., (v_n, v_1)$ are the edges in the circuit.

Differences between Path and Circuit:

1. Closed vs. Open: A path is an open sequence of vertices, while a circuit is a closed loop of vertices.

2. Starting and Ending Points: A path can start and end at different vertices, but a circuit always starts and ends at the same vertex.

3. Cycles: A circuit contains at least one cycle, which is a path that starts and ends at the same vertex. A path does not contain any cycles.

4. Applications: Paths are commonly used to find the shortest route between two vertices in a graph (e.g., using Dijkstra's algorithm), while circuits are often used to identify Hamiltonian cycles (cycles that visit every vertex exactly once) and Eulerian circuits (cycles that visit every edge exactly once).

In summary, a path is a sequence of vertices connected by edges, without any vertex repeated, while a circuit is a closed path that starts and ends at the same vertex.