It's surprisingly simple, and you can understand this even if the last couple sections went a bit over your head. :)

A signed graph is just an ordinary graph with each of its edges labeled with either a + or a -. A polygon in a signed graph is positive if you get + when you multiply all the signs of its edges together, and negative otherwise. For example, a polygon consisting of three positive edges and three negative edges is negative. A subgraph of a signed graph is said to be balanced if all its polygons are positive, unbalanced if it's not balanced, and contrabalanced if all its polygons are negative. Not too bad, eh? Those of you who are really quick may have noticed that this labeling is basically just labeling each edge with an element of GF(2). In fact, there's an extension of the concept of signed graphs, called gain graphs, which orient each edge and assign to each edge an element of some arbitrary group. Gain graphs (and biased graphs, which are a further abstraction of signed graphs and gain graphs) are a bit beyond what I want to talk about here; just suffice it to say that gain graphs have a good number of applications to networking and electrical engineering.

Recall that a graph has a matroid associated with it: it is the matroid for which the ground set E is the set of edges of the graph, and the circuits (minimal dependent sets) of the matroid are just the polygons of the matroid, so that the independent sets are the sets of edges which contain no polygons. For a signed graph (and in general, for every gain or biased graph) there are actually three different matroids which we can associate with the graph, called the bias matroid, the lift matroid, and the complete lift matroid. I'll only talk about the first of these, and only in the context of signed graphs. The bias matroid of a signed graph has as its base set E the edges of the signed graph, and it has the following three types of circuits:

• Balanced (i.e., positive) polygons
• A pair of unbalanced polygons which share exactly one vertex between them (a "tight handcuff")
• A pair of vertex-disjoint unbalanced polygons along with a path that connects them, regardless of the sign of the connecting path (a "loose handcuff").