The excluded 3-minors for Vf-safe Delta-matroids

Bonin, J.E. and Chun, C. and Noble, Steven (2018) The excluded 3-minors for Vf-safe Delta-matroids. Technical Report. Birkbeck, University of London, London, UK.

Abstract

Vf-safe delta-matroids have the desirable property of behaving well under certain duality operations. Several important classes of delta-matroids are known to be vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vf-safe delta-matroids and ribbon-graphic deltamatroids by finding the minimal obstructions, called 3-minors, to belonging to the class. We find the unique (up to twisted duality) excluded 3-minor within the class of set systems for the class of vf-safe delta-matroids. Geelen and Oum [17] found the 166 (up to twists) excluded minors for ribbon-graphic delta-matroids. By translating Bouchet’s characterization of circle graphs to the language of 3-minors, we show that this class can also be characterized amongst delta-matroids by a set of three excluded 3-minors up to twisted duality.

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