Hello and welcome to the next lesson of our course, "Solving Algorithmic Problems with DFS." In previous lessons, we introduced Graphs and their representations, venturing into one of the most predominant procedures in Graph theory — Depth-First Search. Today, we'll learn how to use Depth-First Search to solve a common problem: determining if there is a cycle in the graph.

Detecting cycles in a graph is a common problem that has applications in various domains. It is particularly useful in network routing, deadlock detection in operating systems, and in mathematical problems such as finding the presence of loops in a mathematical sequence.

In a graph, a cycle exists if we can start at a node, traverse along the edges, and return to the same node without retracing any edge. Our task is to construct a DFS function has_cycle(graph), which will check for a cycle in the given graph and return true if a cycle is found and false if the graph is acyclic.

Graphs can be represented in multiple ways, but for this lesson, we will use adjacency list representation for simplicity and efficiency.

We traverse the graph starting from an initial node, and for every visited node, we check if it is being revisited during the DFS exploration. If it is, then a cycle has been detected. Hence, we return true. If no cycle is found after exploring all the nodes, we return false. This approach has a linear time complexity of $O(V + E)$, where is the number of vertices (nodes) and is the number of edges in the graph.