We study the concept of an H-partition of the vertex set of agraph G, which includes all vertex partitioning problems intofour parts which we require to be nonempty with only externalconstraints according to the structure of a model graph H, withthe exception of two cases, one that has already been classifiedas polynomial, and the other one remains unclassified. In thecontext of more general vertex-partition problems, the problemsaddressed in this paper have these properties: non-list, 4-part,external constraints only (no internal constraints), each partnon-empty. We describe tools that yield for each problemconsidered in this paper a simple and low complexitypolynomial-time algorithm.
