Based on Graph

A graph-based label needs a model before an algorithm. Ask what each vertex and edge represents, whether edges are directed or weighted, whether negative weights are possible, and whether the task asks for reachability, shortest path, ordering, or component count. Use an adjacency list for sparse graphs, giving O(V plus E) storage. BFS produces shortest path lengths only for unweighted or unit-weight graphs because vertices leave the queue in nondecreasing distance. DFS is suitable for reachability, components, and cycle detection with visiting states. For directed prerequisite order, use topological sorting and report a cycle when fewer than V vertices are output. Test isolated vertices, disconnected components, self-loops, parallel edges, and a path that begins or ends at the boundary of input IDs. Do not claim Dijkstra handles negative weights. BFS or DFS is O(V plus E) time; Dijkstra with a binary heap is O((V plus E) log V).