Which of the following algorithms can be used to most efficiently determine the presence of a cycle in a graph?

To efficiently determine the presence of a cycle in a graph, we can use the Depth-First Search (DFS) algorithm. Here are the steps:

1. Start by selecting a starting vertex in the graph.
2. Mark the starting vertex as visited.
3. Explore the adjacent vertices of the current vertex.
4. If an adjacent vertex is already visited and is not the parent of the current vertex, then a cycle is present in the graph.
5. If an adjacent vertex is not visited, recursively apply steps 2-4 for that vertex.
6. Repeat steps 3-5 until all vertices have been visited or a cycle is found.

The DFS algorithm efficiently detects cycles in a graph by exploring the vertices in a depth-first manner. It keeps track of visited vertices and backtracks whenever it encounters a visited vertex that is not the parent of the current vertex. This approach allows us to detect cycles in a graph with a time complexity of O(V + E), where V is the number of vertices and E is the number of edges in the graph.