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Parameterized Algorithms for Finding Square Roots

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Abstract

We show that the following two problems are fixed-parameter tractable with parameter \(k\): testing whether a connected \(n\)-vertex graph with \(m\) edges has a square root with at most \(n-1+k\) edges and testing whether such a graph has a square root with at least \(m-k\) edges. Our first result implies that squares of graphs obtained from trees by adding at most \(k\) edges can be recognized in polynomial time for every fixed \(k\ge 0\); previously this result was known only for \(k=0\). Our second result is equivalent to stating that deciding whether a graph can be modified into a square root of itself by at most \(k\) edge deletions is fixed-parameter tractable with parameter \(k\).

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Notes

  1. We restrict ourselves to connected graphs for simplicity. We may do this for the following reason. For disconnected \(n\)-vertex graphs with \(\ell \ge 2\) connected components the natural parameter is \(k=s-(n-\ell )\) instead of \(k=s-(n-1)\). Our FPT result for connected graphs immediately carries over to disconnected graphs if we choose as parameter \(k=s-(n-\ell )\) instead.

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Acknowledgments

We would like to thank two anonymous reviewers for a number of helpful comments that improved our presentation.

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Authors and Affiliations

  1. Laboratoire d’Informatique Théorique et Appliquée, Université de Lorraine, 57045, Metz Cedex 01, France

    Manfred Cochefert & Dieter Kratsch

  2. CReSTIC, Université de Reims Champagne-Ardenne, 7 Boulevard Jean Delautre, 08005, Charleville-Méziéres, France

    Jean-François Couturier

  3. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Petr A. Golovach

  4. Science Laboratories, School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK

    Daniël Paulusma

Authors
  1. Manfred Cochefert
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  2. Jean-François Couturier
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  3. Petr A. Golovach
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  4. Dieter Kratsch
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  5. Daniël Paulusma
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Corresponding author

Correspondence to Petr A. Golovach.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 267959. The research also has been supported by EPSRC (EP/G043434/1) and ANR Blanc AGAPE (ANR-09-BLAN-0159-03). A preliminary version of this paper appeared as an extended abstract in the proceedings of WG 2013 [4].

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Cochefert, M., Couturier, JF., Golovach, P.A. et al. Parameterized Algorithms for Finding Square Roots. Algorithmica 74, 602–629 (2016). https://doi.org/10.1007/s00453-014-9967-4

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  • DOI: https://doi.org/10.1007/s00453-014-9967-4

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