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- W4289671105 abstract "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 $Pi_A$ and $Pi_B$, respectively. This so-called $(Pi_A,Pi_B)$-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 $(Pi_A,Pi_B)$-Recognition problems admit polynomial kernels. In our study, we focus on the first level above triviality, where $Pi_A$ is the set of $P_3$-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph $G[A]$, and $Pi_B$ is characterized by a set $mathcal{H}$ of connected forbidden induced subgraphs. We prove that, under the assumption that NP is not a subset of coNP/poly, textsc{$(Pi_A,Pi_B)$-Recognition} admits a polynomial kernel if and only if $mathcal{H}$ contains a graph with at most $2$ 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 $(Pi_A,Pi_B)$-Recognition, as well as several other problems." @default.
- W4289671105 created "2022-08-03" @default.
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- W4289671105 date "2018-08-27" @default.
- W4289671105 modified "2023-09-27" @default.
- W4289671105 title "Solving Partition Problems Almost Always Requires Pushing Many Vertices Around" @default.
- W4289671105 doi "https://doi.org/10.4230/lipics.esa.2018.51" @default.
- W4289671105 hasPublicationYear "2018" @default.
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