A subgraph or edge set whose circles are all in B (and which contains no half-edges) is called balanced.
Biased graphs are interesting mostly because of their matroids, but also because of their connection with multiary quasigroups.
A minor of a biased graph Ω = (G, B) is the result of any sequence of taking subgraphs and contracting edge sets.
A subgraph of Ω consists of a subgraph H of the underlying graph G, with balanced circle class consisting of those balanced circles that are in H. The deletion of an edge set S, written Ω − S, is the subgraph with all vertices and all edges except those of S. Contraction of Ω is relatively complicated.
Two other kinds are a pair of unbalanced circles together with a connecting simple path, such that the two circles are either disjoint (then the connecting path has one end in common with each circle and is otherwise disjoint from both) or share just a single common vertex (in this case the connecting path is that single vertex).
The fourth kind of circuit is a theta graph in which every circle is unbalanced.
The frame matroid of a biased 2Cn (see Examples, above) which has no balanced digons is called a swirl.
The lift matroid of a 2Cn (see Examples, above) which has no balanced digons is called a spike.
Just as a group expansion of a complete graph Kn encodes the group (see Dowling geometry), its combinatorial analog expanding a simple cycle of length n + 1 encodes an n-ary (multiary) quasigroup.
It is possible to prove theorems about multiary quasigroups by means of biased graphs (Zaslavsky, t.a.)