Graphs

Graphs generalize trees by allowing arbitrary connections and cycles. The two core traversal techniques, DFS and BFS, work on graphs the same way they do on trees, but you must now track visited nodes explicitly to avoid infinite loops. This stage covers the main graph problem types: counting connected components, detecting cycles, finding shortest paths in unweighted graphs, and topological ordering of dependencies. Grids are also implicit graphs, where each cell is a node and adjacency is defined by its four neighbors. Most graph problems reduce to one of these patterns once you recognize the structure.