Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Abstract

A fundamental graph problem is to recognize whether the vertex set of a graph can be bipartitioned into sets and such that ⁡[] and ⁡[] satisfy properties Π and Π , respectively. This so-called (Π,Π) -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,Π) -Recognition problems admit polynomial kernels. In our study, we focus on the first level above triviality, where Π is the set of 3 -free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph ⁡[] , and Π is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that ⁢⊈⁢⁢⁢/⁢⁢⁢ , (Π,Π) -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,Π) -Recognition, as well as several other problems.