On graph contractions and induced minors

Pim Van 'T Hof, Marcin Kamiński, Daniël Paulusma, Stefan Szeider, Dimitrios M. Thilikos

Research output: Contribution to journalArticlepeer-review


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 | VH| 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.

Original languageEnglish (US)
Pages (from-to)799-809
Number of pages11
JournalDiscrete Applied Mathematics
Issue number6
StatePublished - Apr 2012


  • Graph contraction
  • Graph induced minor
  • Graph minor

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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