An example of a problem that can be solved using the depth-first search (DFS) algorithmic approach is finding a path or cycle in a graph.

  The DFS algorithm starts from a chosen source vertex and explores as far as possible along each branch before backtracking. It uses a stack to keep track of the vertices that need to be visited.

  Let's consider a simple example of a graph representing a maze. The maze consists of cells, some of which may be blocked walls. We want to find a path from the starting cell to a specific target cell.

  We can use DFS to solve this problem by modeling the maze as a graph, where each cell is a node and there is an edge between adjacent cells that are not separated by a wall. The starting cell is the source vertex and the target cell is the goal.

  Starting from the source vertex, we mark it as visited and push it onto the stack. We then enter a loop where we pop a node from the stack and examine its neighboring cells. For each unvisited neighboring cell, we mark it as visited, push it onto the stack, and continue exploring from that cell. We repeat this process until we reach the target cell or the stack becomes empty.

  If we reach the target cell, we have found a path from the starting cell to the target cell. If the stack becomes empty before we reach the target, it means there is no path from the starting cell to the target.

  DFS explores the graph deeply before backtracking, which makes it suitable for finding paths or cycles in graphs. It can also be used to perform topological sorting, find connected components, or solve other graph-related problems.