The two fundamental examples in Whitney's seminal paper on independence theory (Whitney 1935) were vector matroids and graphic matroids. It is therefore not surprising that so many aspects of matroid theory are extensions and developments of concepts that were originally introduced in vector spaces or in graphs. In this chapter we shall study in detail the class of graphic matroids. The first section will consider the polygon and bond matroids of a graph, that is, those matroids on the edge-set of a graph Г whose circuits are respectively the circuits and the bonds of Г. The main result of Section 6.1 characterizes those graphs whose bond matroids are graphic. The polygon and bond matroids are not the only matroids that can be defined on the edge-set of a graph. Several other such matroids will be considered in detail in a chapter on matroidal families in a later volume.

In Section 6.2 we shall consider the concept of connectivity in matroid theory, indicating its relationship to various notions of connectivity for graphs. These ideas will be used in Section 6.3, where the main result of this chapter is proved. This result, due to Whitney (1933), characterizes precisely when two graphs have isomorphic polygon matroids. In the final section we shall investigate two graph-theoretic operations having their origins in electrical-network theory. One of the important aspects of these operations in the present context is the fact that they have been very successfully generalized to matroids.