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

TL;DR

This paper characterizes vf-safe delta-matroids and ribbon-graphic delta-matroids by identifying minimal obstructions called excluded 3-minors, and explores their relation to binary delta-matroids and graph minors.

Contribution

It provides a complete characterization of vf-safe delta-matroids through the identification of excluded 3-minors, including the unique minimal obstruction.

Findings

Identified the unique excluded 3-minor for vf-safe delta-matroids.

Established the equivalence between 3-minors and vertex minors in graphs.

Connected the theory of delta-matroids to well-known graph minor results.

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.