Solving partition problems almost always requires pushing many vertices around

A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Π A and Π B, respectively. This so-called (Π A,Π B)-Recognition problem generalizes, amongst others, the recognition of 3-colorable, bipartite, split, and monopolar graphs. In this paper, we study whether certain fixed-parameter tractable (Π A,Π B)-Recognition problems admit polynomial kernels. In our study, we focus on the first level above triviality, where Π A is the set of P3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Π B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP ⊆ coNP/poly, (Π A,Π B)-Recognition admits a polynomial kernel if and only if H contains a graph with at most two vertices. In both the kernelization and the lower bound results, we exploit the properties of a pushing process, which is an algorithmic technique used recently by Heggerness et al. and by Kanj et al. to obtain fixed-parameter algorithms for many cases of (Π A,Π B)-Recognition, as well as several other problems.