On Contracting Graphs to Fixed Pattern Graphs

  • Pim van ’t Hof
  • Marcin Kamiński
  • Daniël Paulusma
  • Stefan Szeider
  • Dimitrios M. Thilikos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5901)


For a fixed graph H, the H-Contractibility problem asks if a graph is H-contractible, i.e., can be transformed into H via a series of edge contractions. The computational complexity classification of this problem is still open. So far, H has a dominating vertex in all cases known to be polynomially solvable, 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 is polynomially solvable. Furthermore, 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. 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Pim van ’t Hof
    • 1
  • Marcin Kamiński
    • 2
  • Daniël Paulusma
    • 1
  • Stefan Szeider
    • 1
  • Dimitrios M. Thilikos
    • 3
  1. 1.School of Engineering and Computing SciencesUniversity of Durham, Science LaboratoriesDurhamEngland
  2. 2.Computer Science DepartmentUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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