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 \(G_1\) and \(G_2\) if \(G\) is a graph of minimum size such that \(G\) is both \(G_1\)-decomposable and \(G_2\)-decomposable.
It is shown that two graphs can have an arbitrarily large number of least common multiples.
All graphs \(G\) for which \(G\) and \(P_3\) (and \(G\) and \(2K_2\)) have a unique least common multiple are characterized.
It is also shown that two stars \(K_{1,r}\) and \(K_{1,s}\) have a unique least common multiple if and only if \(r\) and \(s\) are not relatively prime.