The curse of connectivity: t-total vertex (edge) cover

  • Authors:

  • Henning Fernau;Fedor V. Fomin;Geevarghese Philip;Saket Saurabh

  • Affiliations:

  • Universität Trier, Abteilung Informatik, Trier, Germany;Department of Informatics, University of Bergen, Bergen, Norway;The Institute of Mathematical Sciences, Taramani, Chennai, India;The Institute of Mathematical Sciences, Taramani, Chennai, India

  • Venue:
  • COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
  • Year:
  • 2010

  • Kernels for packing and covering problems

    FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management

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Abstract

We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely k-VERTEX COVER and k-EDGE COVER. Specifically, we impose the additional requirement that each connected component of a solution have at least t vertices (resp. edges from the solution), and call the problem t-TOTAL VERTEX COVER (resp. t-TOTAL EDGE COVER). We show that - both problems remain fixed-parameter tractable with these restrictions, with running times of the form O* (ck) for some constant c 0 in each case; - for every t ≥ 2, t-TOTAL VERTEX COVER has no polynomial kernel unless the Polynomial Hierarchy collapses to the third level; - for every t ≥ 2, t-TOTAL EDG3E COVER has a linear vertex kernel of size t+1/tk.