  We investigate the uniqueness of factorisation of possibly disconnected
finite graphs with respect to the Cartesian, the strong and the direct product.
It is proved that if a graph has $n$ connected components, where $n$ is prime,
or $n=1,4,8,9$, and satisfies some additional conditions, it factors uniquely
under the given products. If, on the contrary, $n=6$ or 10, all cases of
nonunique factorisation are described precisely.
