Bfs alone can find single source shortest path in undirected unweighted connected graph.

BFS and DFS can check if the graph is bipartite or not,find cycle in static graphs,check whether a node is reachable or not ,can check if graph is connected or not,and can find no of connected components

DFS alone can find cut nodes of a graph.

If a topological sort exists then no back edge is formed in DFS.

Dfs can alone check if the given graph is strongly connected or not