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arXiv · 1807.01376

The excluded 3-minors for vf-safe delta-matroids

Joseph E Bonin·Carolyn Chun·Steven D Noble
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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 delta-matroids by finding the minimal obstructions, called excluded 3-minors, to membership in 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. In the literature, binary delta-matroids appear in many different guises, with appropriate notions of minor operations equivalent to that of $3$-minors, perhaps most notably as graphs with vertex minors. We give a direct explanation of this equivalence and show that some well-known results may be expressed in terms of $3$-minors.

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Joseph E Bonin, Carolyn Chun, Steven D Noble. 2019-02-14. The excluded 3-minors for vf-safe delta-matroids. https://doi.org/10.1016/j.aam.2019.04.006

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