SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 640-681, January 2020.
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 ${NP} \not\subseteq {coNP}/{poly}$, $(\Pi_A,\Pi_B)$-Recognition admits a polynomial kernel if and only if $\mathcal{H}$ 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 $(\Pi_A,\Pi_B)$-Recognition, as well as several other problems.

中文翻译：

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解决分区问题几乎总是需要绕过许多顶点

SIAM离散数学杂志，第34卷，第1期，第640-681页，2020年1月。
一个基本的图问题是识别图$ G $的顶点集是否可以分为集合$ A $和$ B $，从而使$ G [A] $和$ G [B] $满足属性$ \ Pi_A $和$ \ Pi_B $。这个所谓的$（\ Pi_A，\ Pi_B）$-识别问题除其他外，概括了对3色图，二分图，分裂图和单极图的识别。在本文中，我们研究了某些固定参数易处理的$（\ Pi_A，\ Pi_B）$-识别问题是否允许多项式核。在我们的研究中，我们专注于琐碎性之上的第一层，其中$ \ Pi_A $是无$ P_3 $的图的集合（集团的不相交并集或聚类图），参数是聚类图中的聚类数$ G [A] $和$ \ Pi_B $的特征是一组相连的禁止诱导子图$ \ mathcal {H} $。我们证明 假设$ {NP} \ not \ subseteq {coNP} / {poly} $，$（\ Pi_A，\ Pi_B）$-Recognition仅在$ \ mathcal {H} $包含图的情况下才接受多项式内核最多有两个顶点。在核化和下限结果中，我们都利用了推入过程的属性，该过程是Heggerness等人最近使用的一种算法技术。和Kanj等人。以获得针对$（\ Pi_A，\ Pi_B）$-Recognition的许多情况的固定参数算法，以及其他一些问题。