If H is a family of graphs, then an H-decomposition of a graph G is a partition of the edges of G each element of which induces a copy of a graph in G.

This paper addresses the problem of finding the number φ(n,H), the smallest number k such that every graph on n vertices has an H-decomposition with at most k elements. φ(n,H) is found in the cases when H={K2,C5}; when H={K2,C5+e}, where e is a chord of the C5; when H={K2,K4−e}; and when H={K2,K3+e}, where e is a pendant edge added to one vertex in the K3.