Minimization and parameterized variants of vertex partition problems on graphs

Let Π1, Π2, . . ., Πc be graph properties for a fixed integer c. Then, (Π1, Π2, . . ., Πc)-Partition is the problem of asking whether the vertex set of a given graph can be partitioned into c subsets V1, V2, . . ., Vc such that the subgraph induced by Vi satisfies the graph property Πi for every i ∈ {1, 2, . . ., c}. Minimization and parameterized variants of (Π1, Π2, . . ., Πc)-Partition have been studied for several specific graph properties, where the size of the vertex subset V1 satisfying Π1 is minimized or taken as a parameter. In this paper, we first show that the minimization variant is hard to approximate for any nontrivial additive hereditary graph properties, unless c = 2 and both Π1 and Π2 are classes of edgeless graphs. We then give FPT algorithms for the parameterized variant when restricted to the case where c = 2, Π1 is a hereditary graph property, and Π2 is the class of acyclic graphs.