Graph

Begin by turning the story into a graph: what is a vertex, what is an edge, whether edges are directed, weighted, or can repeat, and what answer is needed. An adjacency list is normally the right sparse representation, using O(V+E) storage; an adjacency matrix is appropriate only for dense graphs or constant-time edge checks. Use BFS for unweighted shortest path and level order because the queue discovers nodes by nondecreasing distance. Use DFS for reachability, components, and cycle detection with a visiting state. Use Dijkstra only when weights are nonnegative, and topological sorting only for directed acyclic dependency order. Traverse from every unvisited vertex when the task concerns all components. Test isolated vertices, self-loops, parallel edges, cycles, and disconnected input. BFS and DFS are O(V+E) time; binary-heap Dijkstra is O((V+E) log V). A category label is not a complete specification, so ask before implementation.