Biased graphs. II. The three matroids

A biased graph Ω consists of a graph Γ and a class B of circles (simple, closed paths) in Γ, called balanced circles, such that no theta subgraph contains exactly two balanced circles. The bias matroid G(Ω) is a finitary matroid on the edge set E of Ω whose circuits are the balanced circles and the minimal connected edge sets of cyclomatic number two containing no balanced circle. We prove that these circuits define a matroid and we establish cryptomorphic definitions and other properties. Another finitary matroid on E, the lift matroid L(Ω), and its one-point extension the complete lift matroid L0(Ω), are obtained from the abstract matroid lift construction applied to the graphic matroid G(Γ) and the class B. The circuits of L(Ω) are the balanced circles and the minimal edge sets of cyclomatic number two (not necessarily connected) containing no balanced circle. We develop cryptomorphisms and other properties of L0(Ω) and L(Ω). There is no completely general construction rule, besides the bias and lift constructions, which assigns to each biased graph a matroid intermediate (in the sense of independent sets) between G and L and which respects subgraphs. G(Ω) has an infinitary analog for infinite graphs generalizing Klee's infinitary bicircular matroid and the Bean-Higgs infinitary graphic matroid. Whether L(Ω) has an infinitary analog is unclear.