On graph contractions and induced minors

  • Authors:

  • Pim van 't Hof;Marcin Kamiński;Daniël Paulusma;Stefan Szeider;Dimitrios M. Thilikos

  • Affiliations:

  • School of Engineering and Computing Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom;Computer Science Department, Université Libre de Bruxelles, Boulevard du Triomphe CP212, B-1050 Brussels, Belgium;School of Engineering and Computing Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom;Institute of Information Systems, TU Vienna, Favoritenstraíe 9-11, A-1040 Vienna, Austria;Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, GR15784 Athens, Greece

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

The Induced Minor Containment problem takes as input two graphs G and H, and asks whether G has H as an induced minor. We show that this problem is fixed parameter tractable in |V"H| if G belongs to any nontrivial minor-closed graph class and H is a planar graph. For a fixed graph H, the H-Contractibility problem is to decide whether a graph can be contracted to H. The computational complexity classification of this problem is still open. So far, H has a dominating vertex in all cases known to be solvable in polynomial time, whereas H does not have such a vertex in all cases known to be NP-complete. Here, we present a class of graphs H with a dominating vertex for which H-Contractibility is NP-complete. We also present a new class of graphs H for which H-Contractibility can be solved in polynomial time. Finally, we study the (H,v)-Contractibility problem, where v is a vertex of H. The input of this problem is a graph G and an integer k, and the question is whether G is H-contractible such that the ''bag'' of G corresponding to v contains at least k vertices. We show that this problem is NP-complete whenever H is connected and v is not a dominating vertex of H.