A generalization of edge cover
This problem is known to be NP-complete (since the 1970s).
Dyer and Frieze established its NP-hardness even for the highly restricted special case where the graph $G$ is planar and bipartite:


  Martin E. Dyer, Alan M. Frieze:
  On the complexity of partitioning graphs into connected subgraphs.
  Discrete Applied Mathematics 10(2): 139-153 (1985)