On graphs with a unique least common multiple

A graph G is H-decomposable if G can be decomposed into graphs, each of which is isomorphic to H. A graph G without isolated vertices is a least common multiple of two graphs G1 and G2 if G is a graph of minimum size such that G is both G1-decomposable and G2-decomposable. It is shown that two graphs can have an arbitrarily large number of least common multiples. All graphs G for which G and P3 (and G and 2K2) have a unique least common multiple are characterized. It is also shown that two stars K1,r and K1,s have a unique least common multiple if and only if r and s are not relatively prime.