Decomposing subcubic graphs into claws, paths or triangles

Let S = { K 1 , 3 , K 3 , P 4 } be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph G into graphs taken from any nonempty S ' subset of S. The problem is known to be NP-complete for any possible choice of S ' in general graphs. In this paper, we assume that the input graph is subcubic (i.e., all its vertices have degree at most 3), and study the computational complexity of the problem of partitioning its edge set for any choice of S '. We identify all polynomial and NP-complete problems in that setting.