A generalization of edge cover Suppose we are given a general (connected) undirected graph $G = (V, E)$. An EDGE COVER asks a set $S\subseteq E$ of the minimum number of edges, such that each vertex $v\in V$ is incident to at least one edge $s\in S$. Note that this is solvable in polynomial time.

Now I'd like to generalize this problem to find a set $S$ of vertex-induced connected subgraphs that covers each vertex $v\in V$, but each subgraph $s\in S$ consists of at most (or exactly) 3 vertices (The goal is still to minimize $|S|$). Of course, the topology of a connected subgraph with 3 vertices is fixed (the only way is to form a line with 2 edges connecting them). In other words, we need to find a minimum path cover but the length of each path is at most (or exactly) 3.

Is this problem still poly-time solvable or it's NP-hard? Is it related with 3d matching or steiner tree?
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)