Page "Biased graph" Paragraph 18

The frame matroid ( sometimes called bias matroid ) of a biased graph, M ( Ω ), ( Zaslavsky, 1989 ) has for its ground set the edge set E. An edge set is independent if each component contains either no circles or just one circle, which is unbalanced.

( In matroid theory a half-edge acts like an unbalanced loop and a loose edge acts like a balanced loop.

) M ( Ω ) is a frame matroid in the abstract sense, meaning that it is a submatroid of a matroid in which, for at least one basis, the set of lines generated by pairs of basis elements covers the whole matroid.

Conversely, every abstract frame matroid is the frame matroid of some biased graph.