Explain the depth-first search algorithm for a graph.

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Explain the depth-first search algorithm for a graph.

The depth-first search (DFS) algorithm is a graph traversal algorithm that explores as far as possible along each branch before backtracking. It starts at a given vertex and explores as far as possible along each branch before backtracking.

The algorithm works as follows:
1. Mark the starting vertex as visited.
2. Explore the adjacent unvisited vertices of the current vertex.
3. If an unvisited adjacent vertex is found, mark it as visited and recursively apply the DFS algorithm to it.
4. If all adjacent vertices have been visited or there are no adjacent vertices, backtrack to the previous vertex.
5. Repeat steps 2-4 until all vertices have been visited.

DFS can be implemented using either a recursive approach or an iterative approach using a stack. The recursive approach is simpler to understand, while the iterative approach is more efficient in terms of space complexity.

DFS is commonly used for various graph-related problems, such as finding connected components, detecting cycles, and solving maze problems.