To detect a cycle in an undirected graph using depth-first search (DFS), you can use a technique called the "coloring method." This method assigns colors to the vertices during the DFS traversal, specifically three colors: white, gray, and black.

  Here's how you can implement the cycle detection using DFS:

  1. Create a list to keep track of the visited vertices and initialize all vertices as white.

  2. For each vertex v in the graph, if it is not visited, perform a depth-first search on that vertex.

  3. During the DFS traversal, mark the current vertex v as gray (visited but not fully processed).

  4. Then, recursively visit all adjacent vertices of v that have not been visited. If an adjacent vertex u is white (unvisited), recursively call the DFS function on u.

  5. If an adjacent vertex u is gray, it means there is a cycle in the graph. Return true to indicate the presence of a cycle.

  6. After visiting all the adjacent vertices of v, mark the vertex v as black (fully processed), indicating that its exploration is complete.

  7. If the DFS traversal is completed without detecting any cycles, return false.

  By using this approach, the DFS algorithm will detect a cycle by checking if it encounters a gray vertex during the traversal. If it does, that means there is a back edge from the current vertex to one of its ancestors, indicating the presence of a cycle.

  It's worth noting that this method assumes the graph is undirected. If the graph is directed, a slightly different technique known as the "back edge method" is used to detect cycles using DFS.