Computing Small Pivot-Minors

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)


A graph G contains a graph H as a pivot-minor if H can be obtained from G by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. However, so far, pivot-minors have only been studied from a structural perspective. We initiate a systematic study into their complexity aspects. We first prove that the Pivot-Minor problem, which asks if a given graph G contains a given graph H as a pivot-minor, is NP-complete. If H is not part of the input, we denote the problem by H-Pivot-Minor. We give a certifying polynomial-time algorithm for H -Pivot-Minor for every graph H with \(|V(H)|\le 4\) except when \(H \in \{K_4,C_3+ P_1,4P_1\}\), via a structural characterization of H-pivot-minor-free graphs in terms of a set \(\mathcal{F}_H\) of minimal forbidden induced subgraphs.


Pivot Edge Vertex Deletion Minimal Forbidden Binary Matroid Induced Subgraph Relation 
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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceDurham UniversityDurhamUK
  2. 2.LIRMM, CNRS, Université de MontpellierMontpellierFrance
  3. 3.Department of Mathematical SciencesKAISTDaejeonSouth Korea
  4. 4.Université Clermont Auvergne, LIMOS, CNRSAubièreFrance
  5. 5.Department of MathematicsIncheon National UniversityIncheonSouth Korea

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