  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.
