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Durham University

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Dabrowski, K.K., Dross, F., Jeong, J., Kanté, M.M., Kwon, O., Oum, S. & Paulusma, D. (2018), Computing small pivot-minors, in Brandstädt, A., Köhler, E. & Meer, K. eds, Lecture Notes in Computer Science 11159: 44th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2018). Cottbus, Germany, Springer, Cham, Switzerland, 125-138.

Author(s) from Durham


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)|≤4|V(H)|≤4 except when H∈{K4,C3+P1,4P1}H∈{K4,C3+P1,4P1}, via a structural characterization of H-pivot-minor-free graphs in terms of a set FHFH of minimal forbidden induced subgraphs.