Detecting Hamiltonian path in a graph
Hamiltonian Path in a DAG is easy to solve: You can find the longest path in $O(|V|+|E|)$ time using the critical path algorithm: https://en.wikipedia.org/wiki/Longest_path_problem#Acyclic_graphs_and_critical_paths.  For unweighted DAGs, this sounds like essentially the same thing as your topological sort.

Hamiltonian Path is NP-hard on digraphs with cycles, and on undirected graphs.