Edge decompositions into two kinds of graphs

Let H be an arbitrary graph and let K 1 , 2 be the 2-edge star. By a { K 1 , 2 , H } -decomposition of a graph G we mean a partition of the edge set of G into subsets inducing subgraphs isomorphic to K 1 , 2 or H . Let J be an arbitrary connected graph of odd size. We show that the problem to decide if an instance graph G has a { K 1 , 2 , H } -decomposition is NP-complete if H has a component of an odd size and H ≠ p K 1 , 2 ∪ q J , where p K 1 , 2 ∪ q J is the disjoint union of p copies of K 1 , 2 and q copies of J . Moreover, we prove polynomiality of this problem for H = q J .