- finding strong and biconnected components
- orienting an undirected (or mixed) graph to make it strongly connected
- finding an odd directed cycle
- checking uniqueness of perfect matchings
- finding a Hamiltonian path in a tournament

The advantage of path-based dfs is simplicity -- the dfs path captures the structure of many problems, without need for the more complicated dfs tree. A corresponding disadvantage is that more intricate problems (for instance planarity testing) require the full power of the dfs tree.

Path-based dfs is a natural concept, dating from 19th century work on threading mazes (Tremaux's algorithm and others).