The question of whether a graph can be partitioned into independent dominating sets, which is the same as having a fall -colouring, is considered. For , it is shown that a graph can be partitioned into three independent dominating sets if and only if the cartesian product can be partitioned into three independent dominating sets. The graph can be replaced by any graph such that there is a mapping , where is a type-II graph homomorphism.
The cartesian product of two trees is considered, as well as the complexity of partitioning a bipartite graph into three independent dominating sets, which is shown to be NP-complete. For other values of , iterated cartesian products are considered, leading to a result that shows for what values of the hypercubes can be partitioned into independent dominating sets.