This question was motivated by a closely related problem An edge partitioning problem on cubic graphs
Input: at most cubic graph ( maximum node degree is 3) $G=(V,E)$, a natural number $k$
Question: Is there a partition of $E$ into subgraphs isomorphic to $K_{1,2}$ and $K_{1,1}$ such that the sum of the orders of the corresponding subgraphs is exactly $k$ ?
Is this problem polynomial time solvable or is it $NP$-complete?
Edit: Changed input graph to at most cubic graphs to avoid triviality.