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On the Parameterized Complexity of Finding Separators with Non-Hereditary Properties

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Abstract

We study the problem of finding small s–t separators that induce graphs having certain properties. It is known that finding a minimum clique s–t separator is polynomial-time solvable (Tarjan in Discrete Math. 55:221–232, 1985), while for example the problems of finding a minimum s–t separator that induces a connected graph or forms an independent set are fixed-parameter tractable when parameterized by the size of the separator (Marx et al. in ACM Trans. Algorithms 9(4): 30, 2013). Motivated by these results, we study properties that generalize cliques, independent sets, and connected graphs, and determine the complexity of finding separators satisfying these properties. We investigate these problems also on bounded-degree graphs. Our results are as follows:

  1. (1)

    Finding a minimum c-connected s–t separator is FPT for c=2 and W[1]-hard for any c≥3.

  2. (2)

    Finding a minimum s–t separator with diameter at most d is W[1]-hard for any d≥2.

  3. (3)

    Finding a minimum r-regular s–t separator is W[1]-hard for any r≥1.

  4. (4)

    For any decidable graph property, finding a minimum s–t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree.

  5. (5)

    Finding a connected s–t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless \(\mathrm{NP} \subseteq \mathrm{coNP/poly}\).

In order to prove (1), we show that the natural c-connected generalization of the well-known Steiner Tree problem is FPT for c=2 and W[1]-hard for any c≥3.

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Acknowledgements

We would like to thank Pål Grønås Drange for fruitful discussions.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Pinar Heggernes, Pim van ’t Hof & Yngve Villanger

  2. Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

    Dániel Marx

  3. The Indian Institute of Science, Bangalore, India

    Neeldhara Misra

Authors
  1. Pinar Heggernes
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  2. Pim van ’t Hof
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  3. Dániel Marx
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  4. Neeldhara Misra
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  5. Yngve Villanger
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Correspondence to Pim van ’t Hof.

Additional information

This work is supported by the Research Council of Norway and by the European Research Council (ERC) grant 280152. An extended abstract of this paper was presented at the 38th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2012) [20].

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Heggernes, P., van ’t Hof, P., Marx, D. et al. On the Parameterized Complexity of Finding Separators with Non-Hereditary Properties. Algorithmica 72, 687–713 (2015). https://doi.org/10.1007/s00453-014-9868-6

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