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.


  1. polynomial kernel
  2. graph partitioning
  3. monopolar graphs

MSC codes

  1. 05C85
  2. 68Q17
  3. 68Q25
  4. 68R10
  5. 68W40

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Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 640 - 681
ISSN (online): 1095-7146


Submitted: 22 January 2019
Accepted: 6 January 2020
Published online: 10 March 2020


  1. polynomial kernel
  2. graph partitioning
  3. monopolar graphs

MSC codes

  1. 05C85
  2. 68Q17
  3. 68Q25
  4. 68R10
  5. 68W40



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : MAGZ, KO 3669/4-1
H2020 European Research Council https://doi.org/10.13039/100010663 : 714704
Research Executive Agency https://doi.org/10.13039/501100000783 : 631163.11
Seventh Framework Programme https://doi.org/10.13039/100011102 : FP7/2007-2013
Israel Science Foundation https://doi.org/10.13039/501100003977 : 551145/14

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