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 ${NP} \not\subseteq {coNP}/{poly}$, $(\Pi_A,\Pi_B)$-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.

Keywords

  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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References

1.
F. N. Abu-Khzam, C. Feghali, and H. Müller, Partitioning a graph into disjoint cliques and a triangle-free graph, Discrete Appl. Math., 190/191 (2015), pp. 1--12.
2.
D. Achlioptas, The complexity of G-free colourability, Discrete Math., 165/166 (1997), pp. 21--30.
3.
H. L. Bodlaender, B. M. P. Jansen, and S. Kratsch, Kernelization lower bounds by cross-composition, SIAM J. Discrete Math., 28 (2014), pp. 277--305, https://doi.org/10.1137/120880240.
4.
M. Bougeret and P. Ochem, The complexity of partitioning into disjoint cliques and a triangle-free graph, Discrete Appl. Math., 217 (2017), pp. 438--445.
5.
H. Broersma, F. V. Fomin, J. Nešetřil, and G. J. Woeginger, More about subcolorings, Computing, 69 (2002), pp. 187--203.
6.
S. Bruckner, F. Hüffner, and C. Komusiewicz, A graph modification approach for finding core-periphery structures in protein interaction networks, Algorithms Mol. Biol., 10 (2015), 16.
7.
Z. A. Chernyak and A. A. Chernyak, About recognizing $(\alpha, \beta)$ classes of polar graphs, Discrete Math., 62 (1986), pp. 133--138.
8.
R. Churchley and J. Huang, On the polarity and monopolarity of graphs, J. Graph Theory, 76 (2014), pp. 138--148.
9.
R. Churchley and J. Huang, Solving partition problems with colour-bipartitions, Graphs Combin., 30 (2014), pp. 353--364.
10.
M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh, Parameterized Algorithms, Springer, New York, 2015.
11.
R. Diestel, Graph Theory, 4th ed., Springer, New York, 2012.
12.
R. G. Downey and M. R. Fellows, Fundamentals of Parameterized Complexity, Texts Comput. Sci., Springer, London, 2013.
13.
E. M. Eschen and X. Wang, Algorithms for unipolar and generalized split graphs, Discrete Appl. Math., 162 (2014), pp. 195--201.
14.
A. Farrugia, Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard, Electron. J. Combin., 11 (2004), R46.
15.
M. R. Fellows, D. Hermelin, F. Rosamond, and S. Vialette, On the parameterized complexity of multiple-interval graph problems, Theoret. Comput. Sci., 410 (2009), pp. 53--61.
16.
J. Fiala, K. Jansen, V. B. Le, and E. Seidel, Graph subcolorings: Complexity and algorithms, SIAM J. Discrete Math., 16 (2003), pp. 635--650, https://doi.org/10.1137/S0895480101395245.
17.
J. Flum and M. Grohe, Parameterized Complexity Theory, Springer, New York, 2006.
18.
S. Foldes and P. L. Hammer, Split graphs, in Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA), Congr. Numer. 19, Utilitas Math., Winnipeg, Manitoba, Canada, 1977, pp. 311--315.
19.
F. Fomin, D. Lokshtanov, S. Saurabh, and M. Zehavi, Kernelization: Theory of Parameterized Preprocessing, Cambridge University Press, Cambridge, UK, 2019.
20.
F. V. Fomin, S. Saurabh, and Y. Villanger, A polynomial kernel for proper interval vertex deletion, SIAM J. Discrete Math., 27 (2013), pp. 1964--1976, https://doi.org/10.1137/12089051X.
21.
J. Gimbel and C. Hartman, Subcolorings and the subchromatic number of a graph, Discrete Math., 272 (2003), pp. 139--154.
22.
P. Heggernes, D. Kratsch, D. Lokshtanov, V. Raman, and S. Saurabh, Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing via iterative localization, Inform. Comput., 231 (2013), pp. 109--116.
23.
I. Kanj, C. Komusiewicz, M. Sorge, and E. J. van Leeuwen, Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs, J. Comput. System Sci., 92 (2018), pp. 22--47.
24.
I. Kanj, C. Komusiewicz, M. Sorge, and E. J. van Leeuwen, Solving partition problems almost always requires pushing many vertices around, in Proceedings of the 26th Annual European Symposium on Algorithms (ESA '18), Leibniz International Proceedings in Informatics (LIPIcs) 112, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Wadern, Germany, 2018, 51.
25.
S. Kolay, F. Panolan, V. Raman, and S. Saurabh, Parameterized algorithms on perfect graphs for deletion to $(r,l)$-graphs, in Proceedings of the 41st MFCS Conference, Leibniz International Proceedings in Informatics (LIPIcs) 58, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Wadern, Germany, 2016, 75.
26.
J. Kratochvíl and I. Schiermeyer, On the computational complexity of $(\mathcal{O}, \mathcal{P})$-partition problems, Discuss. Math. Graph Theory, 17 (1997), pp. 253--258.
27.
S. Kratsch, Polynomial kernelizations for MIN $f^+\pi_1$ and MAX NP, Algorithmica, 63 (2012), pp. 532--550.
28.
V. B. Le and R. Nevries, Complexity and algorithms for recognizing polar and monopolar graphs, Theoret. Comput. Sci., 528 (2014), pp. 1--11.
29.
C. McDiarmid and N. Yolov, Recognition of unipolar and generalised split graphs, Algorithms, 8 (2015), pp. 46--59.
30.
S. Skiena, The Algorithm Design Manual, Springer, New York, 2008.
31.
J. Stacho, On 2-subcolourings of chordal graphs, in Proceedings of the 8th LATIN Conference on Theoretical Informatics, Lecture Notes in Comput. Sci. 4957, Springer, Berlin, 2008, pp. 544--554.
32.
R. I. Tyshkevich and A. A. Chernyak, Algorithms for the canonical decomposition of a graph and recognizing polarity, Izv. Akad. Nauk BSSR Ser. Fiz. Mat. Nauk, 6 (1985), pp. 16--23, in Russian.

Information & Authors

Information

Published In

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

History

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

Keywords

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

MSC codes

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

Authors

Affiliations

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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