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 ΠB, respectively. This so-called (πA, π_B)-RECOGNITION problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (πA, πB)-RECOGNITION, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A′,B′), and pushes appropriate vertices from A' to B' and vice versa to eventually arrive at a correct bipartition. In this paper, we study whether (π_A, π_B)-RECOGNITION problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where πA 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 πB is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP ⊈ coNP/poly, (πA, πB)-RECOGNITION admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process.