Detecting cycles in a graph is an important problem in graph theory, and it can be efficiently solved using Depth-First Search (DFS). In a directed graph, a cycle exists if you can start at a node and follow a sequence of edges that lead back to the same node. In an undirected graph, a cycle is detected when you encounter a visited node that is not the direct parent of the current node during DFS traversal. In TypeScript, you can implement cycle detection by maintaining two arrays: one to track visited nodes and another to track nodes currently in the recursion stack (for directed graphs). As you perform DFS, if you encounter a node that is already in the recursion stack, a cycle has been found. In undirected graphs, a back edge indicates the presence of a cycle. Detecting cycles is crucial in applications such as deadlock detection in operating systems, dependency resolution in package managers, and circuit design. By implementing cycle detection using DFS in TypeScript, you can efficiently solve problems related to graph traversal and ensure the integrity of data structures that rely on acyclic properties.