Decomposing cubic graphs into connected subgraphs of size three

Let S = K1,3,K3, P4 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 non-empty S′ ⊆ 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 cubic, 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, and give graph-theoretic characterisations of S′-decomposable cubic graphs in some cases. © Springer International Publishing Switzerland 2016.