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.Comment: 14 page